I believe you might be asking two different questions . . One is syncing your Treo with a bluetooth connection, and the other is syncing Versamail wirelessly with your email provider.
The good news is, you can store all messages on your card!
The bad news is, your not going to do it with Versamail
You might want to look at
Chatter or Snapper email. I am a big fan of Chatter and it with my Treo have totally changed how I do email! I have 5 accounts set up on my Treo, and use a Summary screen to view all accounts at a glance. Each account is set with its own "Ring" and Vibration so I know which account just received mail without even looking. It also allows you to turn each account on and off. 4 of these accounts are set to IMAP protocol so its very "push" like. From the moment a message is sent to one of my IMAP accounts to the time my Treo recieves it is right at about 30 seconds.