To answer your question, yes the lights, water pump, and furnace should work without batteries, but the process of removing the batteries may have left the converter disconnected from the trailer. If the trailer has a battery disconnect or "kill" switch, it may have to be in the ON position for things to work, even from the converter on shore power.
Another concern would be that the older converters were very poorly regulated and needed the batteries to filter their output. Lights probably don't care, but any delicate electronics might.
"You cannot reason someone out of a position they have not been reasoned into"
Al, K5TAN and Missy, N4RGO WBCCI 1322
2002 Classic 30 Slideout -S/OS #004
2013 Dodge 2500 Laramie 4x4 Megacab Cummins