This is idle speculation, because (thank heaven!!!) Beren's death is a firmly established fact of the canon, not unlike Balrog wings (they are firmly established to not exist.
)
Of course, idle speculation is always fun, since unlike serious debates (eg. Balrog wings), no one is likely to be offended or insulted or driven into the ground through the crushing grip of reason (still waiting for that to happen to the wingers, though....)
Anyways, my opinion...
I think that Luthien and Beren would have lived together happily, for many years, until Beren passed away from extreme old age (say about 90, the same age as his ancestor Beor).
Luthien, distraught, would have then passed westward to Mandos just as before. She would have sang and pleaded with Mandos, just as before. Then Mandos would offer her a choice, also as before. Her choice would be to live in Valinor with her kin or to be permitted a mortal death like Beren.
So as to be with Beren, Luthien would choose mortal death, and a few years later, Valinor would see its only real, true, lasting mortal death. Which would of course deeply sadden and shock the Valar, and maybe have some bearing on their involvement in Morgoth's defeat.
That's my thoughts, anyway.