I don't know when that friend made their research but there is no town in Poland called Bielut, the closest I could find was Białuty. It's possible that it doesn't exist anymore or was incorporated into another town and would therefore not show up on the register. However, I couldn't find any mention of it at all anywhere which is a bit unusual.
Bielut is also really uncommon in Poland (only 24 people with this name) but there's a much more common name Bielat and, not being an expert on names, even I can think of at least one other name where "a" and "u" got confused when transcribed and stuck so that's another possibility.