After a little while spent spinning wheels, we've gotten the MedCIS integration of CAREO back on track.
It turns out that it may have been a minor config oops, where CAREO was directed to use a URL that may have had intermittent (or otherwise badly behaving) DNS, so the XML-RPC requests were barfing all over the place, for various reasons.
I changed that instance of CAREO to use 127.0.0.1 as the host to use for the JUDURL parameter, and all appears fine now. Absolutely weird, since it shouldn't make a bit of difference.