Use an exchange wallet only for buying and selling Bitcoin. Do not store anything in an exchange. Even a hot wallet or a mobile phone wallet is better than an exchange. It least they won't ask you for an ID.
Also, I find that proxies will only increase the suspicion of cryptocurrency exchanges when you use their websites - as they can easily be identified by IP address.
Exchanges will always identify IP because they are very strict with it, as some nations have restrictions on crypto, so that they would be careful not to have anything to do with those nation, but when somebody uses a vpn or proxy that looks suspicious or IP from a restricted nation, there is every tendency that the account would get suspended or freezed and most times not every newbie have that knowledge as some too do not read ToS before they get registered with exchange so it is not advisable to keep assets with exchange and I align with your opinion that OP should use exchange wallet because nobody would require ID for transaction on a personal wallet.