I haven't tried a different SMS client yet. The stock one does have a Delivery Reports toggle in the settings, so I enabled it; the cause code remains (but I am seeing a delivery confirmation mark on my outgoings). I'm not sure if the sporadic sending still occurs.
(Note: while I did have a few failed sends under Jellybean, they were few-and-far between. They also definitely didn't display a status message like this.)
A Google search turned up some dead threads on other sites, and some people who couldn't seem to send SMSes at all - but nothing that quite matches this.
Any ideas, or similar experiences?