Yes, everything will work, but you won't have "12 volt backup power" for the short time until the batteries discharge... In effect, shore power will provide everything needed to operate all the trailer functions except for large, high demand loads (don't open/close slides without the batteries connected)...
The only difference will be that without batteries, the refrigerator will not "switch to gas operation" if the shore power fails. This time of year, no need for the furnace, if you're gone, no need for hot water or water pump for that matter and with shore power failure, no air conditioning, but it doesn't work on battery anyway.
So, the only difference, essentially, if you disconnect the batteries, would be that the refrigerator won't switch over to gas and you "could" come home to a bunch of spoiled food....
As for no COSTCO, any auto parts store like Auto Zone, Advance Auto or O'Riley's can load test the batteries for you. If they're bad, then your next trip to COSTCO, take them with you. If they're good, then start looking at that fan motor as "starting to fail"......
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John
2015 F250 6.7l 4x4
2014 Cougar X Lite 27RKS
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