I think staining it is fine. You probably want to put some Tung oil on
first to hep pop that grain a bit more. I don't think you can put a
water finish over that though.

I would do hand wiped poly finish. Just get some oil based gloss poly.
Mix it 50% with mineral spirits. Apply with a foam brush and wipe down
with a smooth lint free rag. I'd do 2 coats. Then super light hand
sand, or even wet sand with 400 or 600. Apply final coat. Apply wax
with 0000 steel wool and then buf to the sheen you want.

Of course try on samples that are stanied and sanded the same as the
real piece to start.