Proving that timelike and spacelike spacetime intervals are invariant across inertial frames I'm trying to understand the justification for using the Minkowski metric. It's clear to me that it's the natural choice of metric given that spacetime separations denoted by $(-c^2\Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2)$ are invariant across inertial frames of reference. So the next step is to show that spacetime interval is invariant.
It's also clear that lightlike spacetime intervals are all $0$ and hence invariant. In other words, if $\Delta s^2=0$ in one frame, then it's $0$ in all the other IRFs as well.
But it's not obvious to me how to show that spacelike and timelike spacetime intervals are invariant as well (without assuming Lorentz transformation). I've looked at Why does Minkowski space provide an accurate description of flat spacetime? and Physical reasons for metric definition in special relativity, and as good the answers are to those questions, I still didn't find what I was looking for.
Would appreciate any guidance how to prove that spacelike and timelike spacetime intervals must be invariant. [Tried googling for it myself as well, but no luck]
 A: I think you are asking for too much here.  You say that you want to demonstrate that $-c^2\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$ is invariant, but invariance is a meaningless label unless you specify what kinds of transformations you are considering.
From a mathematical standpoint, there would be nothing inconsistent about taking the group of symmetry transformations between reference frames to be $SO(4)$, and taking the spacetime metric to be $\operatorname{diag}(1,1,1,1)$.  To recognize that this is not a good model for the universe we inhabit, we need physical input.
That input comes in the form of Lorentz symmetry.  You mention in a comment

But I'm looking for a different approach in which we don't use the LT. As far as I know, it should be possible. Using isotropy, homogeneity and the principle of relativity, we can conclude that transformation between IRFs is Galilean or Lorentz. To finally conclude that it's indeed Lorentz, I need to show that the metric is Minkowski, and for that I need to show interval invariance.

A Galilean transformation does not generically  preserve the light-like interval, meaning that a light ray moving in one frame has a different speed from a light ray moving in another.  In order for you to have an invariant speed, one must choose the Lorentz transformations, in which the invariant speed enters as a free parameter.
This is the physical input needed. Since we observe that light travels at an invariant speed, then we can immediately conclude (a) that there is an invariant speed in the first place, so the proper symmetry transformations are Lorentz, and (b) that the parameter which appears in the Lorentz transformation is $c$.  From here, the metric must be covariant under Lorentz transformations, which leads you to the Minkowski metric.
A: To perform such a calculation, you have to think on how you can relate  the spatial and time coordinates from one reference frame to the other. The answer is here. Try to think before clicking the link.
