It's possible to reasonably assume that Hooke's law works in the lab frame. One must first ask if Hooke's law is a correct description of the dynamics in a boosted frame, before one can ask "what is the value of k". If Hooke's law no longer works, there is no sense in asking what the value of 'k' is.
I believe we've been over much of this in another thread. If you orient the spring perpendicular to the boost, I haven't run across any problems so far in assuming that Hooke's law still holds. (This doesn't mean I haven't missed something).
If you orient the spring parallel to the boost, there are some definite issues, though they may or may not show up depending on the value of the boost.
Consider, rather than a spring-mass system, a sound wave traveling through a steel bar. Let v_{\mathrm{sound}} be the speed of sound in the bar.
We know that the correct solution in a boosted frame will be given by the relativistic velocity addition formula, i.e. we will have sound waves in the two directions with speeds v+ and v- equal to
v_{+} = \frac{v+v_{\mathrm{sound}}}{1+v \, v_{\mathrm{sound}}/c^2}
v_{-} = \frac{v-v_{\mathrm{sound}}}{1- v \, v_{\mathrm{sound}}/c^2}
When v is small enough so that v_{+} - v = v - v_{-} there may not be any issue with using a linearized version of the wave equation, but I don't see any way to make a linearized wave equation give the correct answer if this condition does not hold. For moderate boosts, this condtion may hold, but eventually an extreme enough boost will cause the condition to fail.
Under extreme boosts one the problem is that sound waves moving in one direction (the + direction) must havae a different speed than sound waves in the other direction (the - direction). I do not believe this is compatible with Hooke's law, which implies that the sound wave must travel at some velocity that is independent of direction. One can probably avoid this issue by making the speed of sound "slow enough", but this is possible only if one restricts the problems being considered, it won't always work.
So I believe that Hooke's law cannot always work - thus one must spend some time in any particular problem considering whether it works or not before even attempting to apply it.
And the closest thing to a correct approach that I know of for relativistic dynamics I've discussed in
https://www.physicsforums.com/showthread.php?t=183382
The full version of the actual PDE's are quite intractable as one will see. Only by linearizing them can one work with them. Unfortunately, non-linear effects can be important in some scenarios, like a steel bar boosted to a high enough velocity.
The simple approach, which I would strongly recommend, is this:
work the problem in the lab frame, then use the Lorentz transform to describe what the results "look like" in any other frame. I.e. do the dynamics in the lab frame, and use kinematical principles to describe how the results work in different frames. The problem is that doing the dynamics requires some approximations to be tractable even with the simplest assumptions, and these approximations which work in the lab frame may not work in the boosted frame.
Assuming that the Newtonian dynamics one used to use are compatible with relativity is not guaranteed to give one a correct answer. One may get lucky and get the right result, one may not be so lucky. This is not the right way to do physics IMO, - the right way involves getting the correct answer without relying on luck.
