Can anyone help identify a (presumably) vintage Battaglin bike for my uncle?
I'm going to go against the grain a bit here, and say that you've got a great example of something from the dark ages of Italian bicycle manufacture. It is an early 90s bike. I know this because the phone number on the shop's sticker starts with an 01, dating it to after BT changed all the area codes.
The welding is dreadful/ugly and while the bike will be quite exciting to ride and is a great colour, they tend to be quite harsh and unforgiving, so you'll know it in your bones after you've been out for a long ride. The aluminium aero fork is completely badass though, really nice, though firm!
Last year (2020) I sold a very similar Coppi that had come from the tip; I valued it at £100 after researching past ebay listings for similar bikes. You can try the same method. The guy that has it now really likes it because it looks cool and it has a famous name on it, but it isn't a valuable bike.
Enjoy it for what it is, a cool looking bike with some Italian flair but missing the craftsmanship associated with the older bikes of the same brand.

I'm just going to add that the Coppi Gavia I'm referring to could actually be based on the exact same frameset as the details look pretty much identical.