Firstly, the ADM mass of an end is in no way the sum of the masses of the things in any portion of (or entirety of) the spacetime. Even the ADM mass of a black hole or even a regular star is in no way the sum of the masses of the parts. For instance if you put 10^20 cold iron atoms into a spherical asteroid in an otherwise empty universe, it will have an ADM mass that is less than 10^20 times the mass of an individual iron atom. The ADM mass is not equal to the sum of "other" masses of any kind.
The ADM mass of an end is just a function of the metric in one of the asymptotically flat regions. So you can choose a region of the manifold that topologically looks like $\mathbb{R}^n-B^n$ and where the metric is close enough to the flat metric in spherical coordinates and then compute the metric out there, and then compute the mass (of the end) from that. It isn't related to any other end, or to the compact region $K$.
If you have a manifold with multiple ends, colloquially you can compactify all the ends except one (in a way that doesn't change the mass of that end) and compute the mass of that remaining end. So when Robert Bartnik says that "the mass of an asymptotically flat n-manifold is a geometric invariant" I read that as that the mass didn't depend on a choice of coordinates in which the end is asymptotically flat. And that the generalization to multiple ends and/or geodesically incomplete manifolds just means that the mass of each end is a separate geometric invariant (in the sense that each end has the mass it has regardless of which coordinate system you use of the potentially many coordinate systems in which it looks asymptotically flat).
As for the wormhole and its relationship to positive mass theorems, in my experience only very few proofs of positive mass theorems explicitly state the assumptions of various energy conditions (and sadly even fewer statements of the theorems themselves explicitly mention these assumptions), but to my knowledge every positive mass theorem assumes them. But more importantly don't let the word mass confuse you, the mass of an end is not the sum of some type of primitive masses added up over a region.
Different ends of the same spacetime can have different masses. The mass of an end is not the sum of any basic mass of stuff inside the/a bounding surface of an end.
And I want to make another thing clear about geodesic (in)completeness and the ADM formalism. The ADM mass is simply a definition about the asymptotic behavior of the metric on an end. The metric has the property (or not) regardless of whether it is geodesically complete. If being geodesically incomplete interferes with your proof or your computation, then simply adjust the manifold to make it complete in a way that doesn't change whether it has a mass or what mass it has, and then move on with your life. As a definition it just depends on the end. And it is a property of the spatial metric, it really has nothing to do with evolution or the ADM formalism (again don't let the name confuse you), the definition is just about the (in space) asymptotic (spatial) metric of an end. You don't need a lapse or a shift to talk about the ADM mass of an end.
Edited to include explicit examples with ends with different masses
Robert Bartnik 100% definitely means what I say rather than what you say, because otherwise he'd be wrong because there are examples of manifolds with multiple ends where the ends have different ADM masses. This simply isn't a matter of opinion. A specific metric with two differently massed ends is at the end of this answer.
I will construct a manifold with two ends, each of which has a different ADM mass. First I will develope background techniques, since a reader might need to remedy many misconceptions before I get to my counterexample.
The metrics in question are spatial metrics (so 3d manifolds when you have a 4d spacetime such as in general relativity). But I'm going to start with some 4d manifolds so you can see the exact connection between the theory and general relativity.
Our first 4d spacetime is the maximal time symmetric Schwarzschild black hole, let's use Kruskal-Szekeres coordinates with metric:
$$\frac{4(2M)^3}{r}e^{-r/(2M)}\left(dR^2-dT^2\right)+r^2\left(d\theta^2+\sin^2\theta d\phi^2\right),$$
with $T^2-R^2=\left(1-\frac{r}{2M}\right)e^{r/(2M)}.$ This has a white hole singularity at $T=-\sqrt{1+R^2}$ and a black hole singularity at $T=+\sqrt{1+R^2}$. Event horizons at $T=R$ and $T=-R$. It is geodesically incomplete. A reader is probably familiar with it (or should become familiar). Note that the region with $R>+\sqrt{T^2}$ is causally unrelated to the region with $R<-\sqrt{T^2}$
Now, take the surface where $T=0$, this is a 3d manifold with metric:
$$\frac{4(2M)^3}{r}e^{-r/(2M)}dR^2+r^2\left(d\theta^2+\sin^2\theta d\phi^2\right),$$
and note that it is geodesically complete, and that the region with $R<0$ is still most definitely causally unrelated to the region with $R>0$. Geodesic completeness is unrelated to causal influence when you talk about spatial sections, and even being geodesically complete in a spatial slice tells you nothing about whether you are geodesically complete as a 4d manifold (after you put in a lapse and a shift if even possible).
So let's keep that example in mind, but let's now develop another basic tool. Imagine a flat region of minkowski space, say $\{(\tau,x,y,z): x^2+y^2+z^2<4\}$ and an asymptotically flat Schwarzschild solution $$-(1-\frac{1}{r})dt^2+(1-\frac{1}{r})^{-1}dr^2+r^2\left(d\theta^2+\sin^2\theta d\phi^2\right) \text{ with } r>2.$$
These two solutions can be sewed together, corresponding to to single solution where there is a source term at $r=2$ of positive energy density. If you don't like the singularness of the source term, smooth it out from $r=2-\epsilon$ to $r=2+\epsilon$. The point is that if you place actual positive energy on a spherical shell, then the effective mass "outside" the shell looks bigger (while the inside doesn't notice except for the time dilation between inside and outside the shell). This is very common, every single star in the entire universe, now or ever, has this property. Every planet, every asteroid, etc.
Yu can imagine that spherical shell moving in time, but for any fixed Schwarzschild/Minkowksi time it is at some radius, so the spatial metric always looks like a Schwarzschild glued to a Minkowski, or a smoothed version. Smoothing a region that has a Schwarzschild of lesser mass on the inside and larger Schwarzschild mass on the outside is nothing difficult and results in a positive energy density for joining, this is exactly what every normal planet or star does.
So now imagine the solution before $$\frac{4(2M)^3}{r}e^{-r/(2M)}dR^2+r^2\left(d\theta^2+\sin^2\theta d\phi^2\right),$$
(recall that $0^2-R^2=\left(1-\frac{r}{2M}\right)e^{r/(2M)}$).
If we place the right amount of mass or energy density at $r=8M$, then we can set up two coordinate system, the first allows $R_1$ to be positive or negative, has $\theta$ and $\phi$ as normal, and has metric:
$$\frac{4(2M)^3}{r_1}e^{-r_1/(2M)}dR_1^2+r_1^2\left(d\theta^2+\sin^2\theta d\phi^2\right).$$
It holds for every negative $R_1$ and for all $R_1$ up until the first place where $r_1=8M$ when $r_1$ is implicitly defined through $0^2-R_1^2=\left(1-\frac{r_1}{2M}\right)e^{r_1/(2M)}$, i.e. for any $R_1<+\sqrt{3e^4}$. For our other coordinate system we have $R_1$ is positive, has $\theta$ and $\phi$ as normal, and has metric:
$$\frac{4(4M)^3}{r_2}e^{-r_2/(4M)}dR_2^2+r_2^2\left(d\theta^2+\sin^2\theta d\phi^2\right).$$
It holds for large positive $R_2$ up until the first place where $r_2=8M$ when $r_2$ is implicitly defined through $0^2-R_2^2=\left(1-\frac{r_2}{4M}\right)e^{r_2/(4M)}$, i.e. for any $R_2>+e^1$.
If you don't like the discontinuity you can instead add a shell of positive energy density near $r=8M$ so that when $r>8M+\epsilon$ you are exterior Schwarzschild with mass $2M$ and such that when $3M<r<8M-\epsilon$ you are exterior Schwarzschild with mass $M$, and just continue that mass $M$ solution all the way tot he other end. If you disagree about this (all in the region $r>3M>2M=r_s$) then you basically disagree that you can make more massive stars by adding positive energy density material in a shell about the star. It has nothing to do with singularities, geodesic completeness or anything, since the exterior to an actual (uncharged non rotating static stationary) star is Schwarzschild.
So the mass $M$ solution goes all the way to the other side but on this side it transitions to a mass $2M$ solution, one ends has mass $M$, the other has mass $2M$. It is a geometric invariant of each end.
You could also imagine adding some mass, $m$, in one shell, then half that mass, $m/2$, at a larger surface area shell and then adding yet another mass, $m/4$, at an even larger surface area shell and so on so that the mass of the end is increased by $2m$. Each time you only affect the outside, since the you can adjust the end without changing the interior at all. So it is definitely the mass of an asymptotic end, not the mass of some volume.
You fixation of thinking it is about a volume when it is about a limit of surface integrals is a problem even with just one end in the almost simplest situation a regular situation of a normal star with layers of normal positive energy density. And I don't know where you get that idea, the definition itself says it is about a limit of surface integrals. Geodesic completeness is unrelated to anything, we can remove the region where we transition from a star of one mass to a solution of a star of the other mass, so the smoothness of the transition doesn't matter. So all that remains is that ends of different masses can indeed be joined together, exactly as I said.
