Your conclusion that
I think a pseudovector is actually a function of vectors equipped with preallocated arguments and every transformation on a vector has a counterpart on a pseudovector like $\tilde P $.
is, I think, essentially correct; the reason parity is confusing is that we tend to drop the tilde.
More precisely, whenever parity considerations are on the table, for all practical purposes vectors and pseudovectors live on different vector spaces, which I'll call $V$ and $W$. One key fact that makes this work is the fact that the addition of a vector and a pseudovector is ill-defined, and such a combination is never used or physically relevant.
In that sense, the cross product is a bilinear function
$$\times:V\times V\to W$$
whose domain is a different vector space. (Similarly, you can define the analogous operations $\times:V\times W\to V$ and $\times:W\times W\to W$, which are never in danger of confusion with each other.) The parity operation then splits into two transformations, $P_V$ on $V$ and $P_W$ on $W$, which are related to the cross product as
P_V(v_1)\times P_V(v_2)=P_W(v_1\times v_2).
However, because you never apply $P_V$ to objects in $W$ and vice versa, it is acceptable to drop the subscripts.
Now, I've been deliberately vague about what $W$ actually is. It is certainly isomorphic to $V$, and shares a good bit of structure with it (for example, you can extend the dot product to combinations from both, though that does get you a pseudoscalar). However, the versions that are useful for generalizations to higher dimensions and more complex geometries (e.g. curved spacetime) look rather different; they are nicely outlined in this answer by ACuriousMind.
In the end, though, I think that questions of the form "what is, mathematically, the object $X$?" are not really that useful; instead, the fruitful questions that really get you forward on the maths are of the form "how does $X$ behave mathematically?". The answer to that is usually a set of axioms which are enough to tipify the object's behaviour, and even uniquely specify it up to a canonical isomorphism. In that case, you can come up with constructions which prove existence, but in a sense all you need are the axioms.
One case where that happens is with tensors, where the universal property is such an axiom. (For examples, see this thread.) As far as pseudovectors go, the facts laid out above are pretty close to a complete set of axioms that constrain their behaviour closely enough to be useful.