So I think I've kludged together a way around this problem.
- Run mbox2eml on my Thunderbird mailbox
- Import resulting e-mail files to Windows Mail (aka Outlook Express)
- Import Outlook Express e-mail to Outlook 2007
- Write a VBA macro in Outlook to open each e-mail, delete all attachments that are HTML copies of the plaintext messages, and finally remove all recipients that aren't my e-mail address, saving each message.
- Import the resulting Outlook mail back into Thunderbird and EIS for Palm
- Sigh.
Sadly this only works if I constantly go import and export my mail, and it doesn't work at all if I check my mail directly on the Visor via dial-up.