In fact iCal use some hidden applescript to send the invitation.
They are located in ical.app/contents/resources/scripts/mail.scpt.
Modify the part to achieve your goal:
on send_mail_sbrp(subjectLine, messageText, myrecipient, myrecipientname, invitationPath)
You need to be very careful. Make a copy of the original mail.scpt prior to any modification.
If you know applescript then you will be able to send it from ThunderBird.
This is I think the only way to do it