This is an interesting idea. I was going to set up an email server on my VPS with just SMTP that auto-forwarded everything to my gmail. But this might be simpler.
Do NOT do this. Unless you're willing to make absolutely sure that no spam gets through to gmail, you will be blacklisted by gmail when it flags spam coming from your host. And not even a lot of it.
Basically, to use gmail this way you have to be almost as good as gmail at the thing that most makes gmail worthwhile. It's not worth it. Much easier to go the other way.
For the time being, you may take advantage of Gmail's POP3/IMAP support to download emails hosted on Pawnmail from Gmail. (See Google's article https://support.google.com/mail/answer/21289?hl=en) The emailed is pulled from Pawnmail rather than pushed, but it achieves the same effect.