Take a look at coinjar, here:
https://www.coinjar.com/Or you could also check
https://www.quadrigacx.com/ This one is a Canadian-based exchange so they're more likely to have the withdrawal option you want.
Ask their support directly.
Also, here's a great list of exchanges :
https://www.bestbitcoinexchange.io/It's always a good idea to shop around, see what fits your needs best.
As for wallets, the online ones I have used are blockchain.info, coinbase, and xapo. Two of which are ruled out, which leaves xapo. You could also ask them.
If you're not comfortable giving out ALL your information, then take the time to look at other options.
And remember to
check everything for yourself.
(Saw some discussion about coinsbank here:
https://bitcointalk.org/index.php?topic=1708586.0)