Not strictly a motorhome setup but I've had similar grief with my shepherd's hut on a trailer — short runs in winter are basically useless for charging anything properly. The alternator barely gets going before you're pulling back onto the drive.
Few things that actually helped me:
- DC-DC charger (B2B) — absolute game changer. Victron Orion-Tr Smart is the one most people recommend and it genuinely works. Pulls a proper regulated charge rather than whatever scraps the alternator feels like chucking at the battery
- Fogstar lithium — switched from a knackered AGM last spring and the difference in how it holds what little charge it gets is night and day
- Solar doing the heavy lifting — even a couple of 100W panels on the roof helps bridge the gap between drives
The real problem with winter short runs is your engine battery is also getting a hammering trying to start in the cold, so the alternator's basically doing triage and your leisure battery is last in the queue.
Honestly if you're mostly doing short local runs with the odd longer trip, I'd just lean harder on solar + a decent charger when you're on hookup rather than expecting the alternator to save you. It won't. It's basically useless in that scenario and will just gaslight you into thinking everything's fine until it very much isn't. 😅
What battery chemistry are you running at the moment? That makes a big difference to how badly you feel the pain.