I am very confused why you even need an exchange for this? Aren't you a trader? Get it through your broker. I'm not referring to just shitty HOOD for millennials here, but a real trading app like IBKR. You have your choice of either buying the futures, or buying the calls through CME Group, or you can grab any of the ETFs, or you can even purchase SPOT value directly through the platform as well. What more do you want with just one brokerage app? And it's the best. * Edit. Ok, I just noticed you also asked about custody. I assume you may refer to self-custody here?
Most Bitcoin ETFs are custodied by Coinbase. So might as well register on Coinbase. You can lose your keys or have them stolen. If Coinbase gets hacked it's game over anyways. And if they get hacked, i guarantee you there would be a hard fork to restore the lost coins. So for most people Coinbase is probably safer than self custody. Coinbase also has around $10B of cash reserves.
True, Coinbase does seem safer with their reserves, but I still prefer having control over my keys for peace of mind!