After almost 2 years on 1 set of batteries, the outside unit stopped sending data. A battery change seemed to solve the problem, but only for a few hours. It stopped sending again the same day. After a lot of messing around, it was apparent that the unit was stopping when the relative humidity went above 88% for a few hours. If the RH reduced, it started again all on its own. The old batteries were down at 1.2V, so it seems that they were the original problem. Perhaps changing the battery cracked the (patchy) varnish coating that was originally applied?
A new covering of varnish seems to have done the trick. Well, 1 day in, but RH reached 99% for a few hours last night.
It's worth noting that the transmitter sends a random station ID on power up. If this code is different to the old one, this can mean that the internal display stays blank for a few hours before realising what's going on. You can reset the display, after downloading the data, if you don't want the wait.