Why would they flag me if I open a bunch of accounts with different IPs using a VPN? How would they know I'm using a VPN?
Using a VPN doesn't mean you are safe and your IP address isn't traceable. Read this article,
Can you be tracked with a VPN?. There is not much VPN provider. So most likely, they have a list or some other way to detect VPN providers. Here is an important part.
The IP addresses of VPN servers aren't difficult to recognize — there are even databases that try to determine whether an IP belongs to a particular provider. When you access a website with a VPN on, it may be able to identify that you're using a VPN using your IP. However, this doesn't mean the website will know the identity of the person behind the IP address — just that they are using a VPN.
It seems although they wouldn't detect who are you but still they are able to know the traffic came from where. In this way, you just risking yourself I think. Binance has withdrawal limits without KYC, you can withdraw 2 BTC per day without verification. So at least you may do it once a day. Or you may look for some trusted person who will exchange for you on behalf of you who have verified account. As far as I know, withdrawal limits for the verified user are 100 BTC.