I think the more ordinary reading of that passage, and the one that leapt to my mind immediately upon first reading it and every time thereafter, is that Merry was experiencing the memories of the man who had died in the battle.

Indeed, the barrow wights appear to be untethered souls who wander that ancient battlefield, which on further examination in the backstory was the ground of one of the most significant battles in the history of Middle Earth. People (men and hobbits alike) are said to steer clear of it because of the dangers those souls present, presumably the same kind of danger Merry experienced.

It’s probably not automatically reasonable that everyone who experienced danger experienced it because they were reincarnated. I mean, I guess you could receive that meaning, but only through a complex knot that Occam’s razor would rather slice through to the more logical “haunting” explanation, an explanation Tom Bombadil himself seemed to warn the hobbits of.

None of this is to discount the themes of reincarnation in the novels, on which point I agree with you, only to note that this one is a bit of a stretch, though not impossible, of course.