I was in the exact situation as you are. I moved my custom domain to iCloud Mail. I'm not sure how complete the custom domain is with iCloud Mail. It basically

I was in the exact situation as you are. I moved my custom domain to iCloud Mail. I'm not sure how complete the custom domain is with iCloud Mail. It basically uses the custom domain email to redirect to your @icloud.com email address. In email clients I have to login as username@icloud.com and set my name@lastname.com email as an alias. It works perfectly though, I'm very satisfied. My Google account still exists at name@lastname.com, I can still OAuth with that account. My Gmail inbox is just not updated anymore since the emails are actually going to Apple's servers. I requested a Google Takeout (basically all the files and emails associated to your account" so if I ever lose access to that account, I don't lose the files.