Mine even after a new set of batteries will still go for a few hours at times in battery saver mode. My OEM batteries died last winter. Confirmed KIA...
I'm worse off than you, 180amp alternator (or whatever the smaller one is). It's not 220amp. I'm thinking of going dual. On one hard winch pull I got battery saver message immediately with no exterior lights on. I had the engine at 2000rpms while winching to boost the amps up immediately after getting the message. Stalled the winch twice but that's not the alternator - even with a shackle double hauling the suv I was trying to tug from a muddy lake in sub zero temps it was just too buried. It was suck by suction and a large chunk of ice behind the axle. If your only using the winch once in awhile and can afford to go a few hours after a hard winch pull and or wait a bit between hauls to make sure your battery has enough for the next start in winter conditions you will be ok. So far this year at -8F I hit battery saver mode a few times for hours at a time running only light cannons and the ARB for only enough to lock front and rear up. Doesn't worry me but makes me want dual alternators. I believe the grid heater is the massive draw the system is seeing and trips it into battery saver most often. Last year I was in battery saver ever morning new batteries or not. But it was an exceptional winter.
Make sure your connections to the battery for accessories are correct and are not inter fearing with the shut system. There are right and wrong terminals particularly on the drivers side I believe. Google the ram manual for the OEM switch panel, it talks about it in that.
IMHO you'll be fine most likely.
I want to go dual just because it can get so cold for so long in my region of the country. Maximum time to recharge the battery would be good since truck is in sub freezing temps from Sept to March without block heater or garage for 50% if it's starts....
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