I own my own domain and when I visit the control panel I can log into the mailbox of any of my email addresses, and then send and receive from there. Is that not the case with yours?
As for being locked out, support have told you what you need to do. I appreciate it may not be what you want to do, for one reason or another, or you feel there may be a better solution… but they have given you a solution. It’s obviously not one you like, but it is one you need to follow to get this resolved.
Wouldn’t it be quicker, easier and less stressful to just email them from your Paypal email address? You’re welcome to carry on questioning the quality of support, and debating the merits of their solution afterwards, but at least you’ll then have the file you paid for.