If it helps any, "Bethlehem" in Polish is "Betlejem", "was sleeping" would be "spał", and "carol" is "kolęda". If you do YouTube searches with these words in combinations, you may hit upon a clip of the carol you want.

The carol "Dzisiaj w Betlejem" (Today in Bethlehem) seems quite popular--in fact I'm sure the tune is used for a Belarusian carol, probably a variant of the same carol.

On the other hand, one description of "Bethlehem lay sleeping" that I found says "Polish carol melody", which may mean that there is no direct relationship between the English and Polish words, in which case searching simply for "kolęda" or "kolędy" (plural) may be most fruitful.