The setup you describe doesn't bring QM and SR into conflict because the two split halves of the wavefunction never do anything interestingly "quantum". The radioactive decay process acts as a measurement of "which branch of the beam-splitter did the particle go down?" and so the experiment can be analyzed as if the particle went down one branch of the other in its entirety and was never split.
To get at the quantum effects you'd have to consider a case where the two parts of the wavefunction are eventually brought back into interference with each other, but any sort of decay process is going to make them mutually incoherent, so you won't see any interference fringes. You might get something interesting by having a moving mirror in an interferometer, which would redshift the reflected wave in that arm by a bit, and that would result in moving interference fringes due to the resulting frequency difference (you'd need to gate the interference pattern on the position/velocity of the mirror to see this.)
But even in that case, nothing very interesting would happen. In any experiment where you can detect which arm a particle goes down it will behave as if it went down that arm, even if your choice is delayed. In any interference experiment it will be impossible to tell which arm it went down. Nature is really good at doing the book-keeping to maintain the quantum veil.
I was guessing that there might be tricking some EPR thing
due to the two paths being different in SR. Or I was
trying to mess up the "bookkeeping"!
You seem to want to get the situation back to the
Michelson-Morley interferometer! Okay! Or, sure,
just Young's double slit.
I know; it's not nice and not easy to fool Mother Nature!
When I get done with software and business, I want
to take a careful pass through SR, GR, and QM looking
for just something wrong with the bookkeeping. Wheeler was, so I'll try too!
To get at the quantum effects you'd have to consider a case where the two parts of the wavefunction are eventually brought back into interference with each other, but any sort of decay process is going to make them mutually incoherent, so you won't see any interference fringes. You might get something interesting by having a moving mirror in an interferometer, which would redshift the reflected wave in that arm by a bit, and that would result in moving interference fringes due to the resulting frequency difference (you'd need to gate the interference pattern on the position/velocity of the mirror to see this.)
But even in that case, nothing very interesting would happen. In any experiment where you can detect which arm a particle goes down it will behave as if it went down that arm, even if your choice is delayed. In any interference experiment it will be impossible to tell which arm it went down. Nature is really good at doing the book-keeping to maintain the quantum veil.