I was just reviewing the Haenel website and they do not have .243 listed as an available caliber choice. Moray, maybe you have access to this caliber in the UK with the Haenel? The only rimless cartridges that I see are 30-06, .308, .270, 300 win, and 7mm Mag. I would, myself, only consider the .308 or .270 and possibly the 30-06 in a single shot as the magnums are pretty brutal from the recoil standpoint...
Another rimmed caliber to consider for your tasks is the 7x57r. This is the rimmed version of the classic "do everything" cartridge that will take everything from foxes to medium range Reds and other larger deer. It is a very effective and very mild cartridge for a break barrel rifle. The 6.5x57r was mentioned as well, but it is much more difficult to find, as a rule.
I do not have any experience, admittedly, with non-rimmed cartridges in single shot guns. I do have some friends who have them and, while not as easy to extract, do not complain about it as being a big issue. .243 (if you can get it) certainly is everywhere and cheap to boot. Good luck with your decision.
Chirug