First of all, the demonstration of Bell inequality violations via experiment has proven that no theory of LOCAL hidden variables can reproduce the prediction of QM. However Bohmian mechanics is a NON-local HV theory, so it is not addressed by Bell's theorem at all.
Bohmian mechanics has been shown to be completely equivalent to standard QM in terms of its ability to reproduce and predict experimental results. I cannot comment on the validity of BM in relativistic regime, however I have seen DeMystifier respond to others that his published research addresses those issues. You might want to search for those responses.
As far as experimental disproof of Bohmian mechanics is concerned, that is a very tall order at this point. BM is an interpretation of QM, in that it addresses what goes on "behind the scenes" ... the reason that this is open to interpretation in the first place is that all of the experimentation that has been done to date in QM has not given us insight to know what actually happens when QM systems are not being measured. So, any experiment that could even reliably test the hypotheses of Bohmian mechanics would also necessarily test the CI, and MWI, and all the other interpretations as well.
It might also cut the traffic on the QM Forum at PF by 50% or more, once the first flurry of discussion ended. Once the question of which interpretation is correct is settled, what will we have to argue about incessantly?