Bitstamp does have pretty ID verification from what I believe and isn't like regular exchanges that just require a passport/drivers license and a proof of address. I wouldn't really use them for anything, especially with so many exchanges out there that are very trusted and also don't require KYC unless you want to trade a lot of funds daily (Binance, Kucoin).

I've heard from friends that they ask for the source of your BTC (for a mining friend, he had to take photos of him next to miners, and showcase a lot of other privacy-breaching stuff just to withdraw his funds), as well as other things mentioned by figmentofmyass.

As much as it sucks, unless you are willing to give out a LOT of information about you, give up. Hopefully, it isn't a lot of money and you recover mate.