Now that 3rd party apps cannot read APN information, it seems that there's no way to send MMS programmatically (unless you ask the user to manually enter APN info -AND- you have a working APN defined in the phone's settings), so the only option seems to be sending the MMS using the stock MMS app.
I've saved an MMS in the native MMS content provider (content://mms/outbox) and it successfully appears in the native app but is not being sent until I manually sent another MMS.
I'm looking for a broadcast, or any action that would trigger sending all messages in the outbox of the native MMS app.