The email you use for the Apple ID does not have to be the same as the one you actually use for email.
You don't need to use your son's existing email as an Apple ID either. You can create a different ID for Apple ID, then initially set his real email as the secondary. This would avoid the issue you are facing. The flow is like this: New disposable email> Register as Apple ID>Real email as secondary> get confirm email from Apple>change real email to primary email.
You need to have email access for the email you associate with your Apple ID in the initial on-boarding steps (you have to confirm that you have access to the email account), but there is no requirement that you use that email address after on-boarding. In the Apple ID configuration panel, you can add additional email addresses, and use those other email addresses as the primary email for contact emails as well as the email you wish to use for things like Messages. This way all emails come to the 'real' email and nothing goes to the 'temporary' email.
For example, I have an old hotmail.com email address that I still have access to, but have abandoned due to excessive spam. I use a gmail.com address instead for daily use and as my 'public' facing email. However, my Apple ID is the hotmail.com address, while Messages, and contact emails and notices from Apple are all sent to the gmail.com address as the 'public' address. I simply 'unchecked' the hotmail.com email as one where I want Messages, etc sent. The only time I ever even have to remember this email address is when I type it as my Apple ID.
A suggestion is to create a secondary email account for your son to use for the on-boarding process, then add his 'real' email to the Apple ID preferences. I would suggest he keep this secondary email but there is no reason to use it. If he has a school email address, this might be a good one to use for the Apple ID, but then his personal email becomes the one that is 'public'.