The quick check would be measure across the battery terminals with your voltmeter with shore power turned off. If the battery is fully charged it will read around 12.6-7VDC. Then turn on the shore power and the same battery terminals should read 13.6VDC telling you the converter/charger is ON and the connections between the converter and the battery are all good.
If you do not see the increase in DC VOLTAGE at the battery terminals when you turn on the shore power then the converter/charger may be bad, or the interconnect fuses between the converter and battery are bad, or you have some sort of bad cable connections or perhaps you have a BATTERY DISCONNECT switch open.
I had to chuckle when i read you was going to pick up a DC Multimeter as they were on sale.... They only cost $7-$20 available from LOWES-WALMART-Any number of autoparts stores. Sometimes you get one free when buying some RV related items from Northern Tools or Harbor freight....
Good luck with tracking down your battery not being charged
just my thoughts
Roy Ken
My Posts are IMHO based on my experiences - Words in CAPS does not mean I am shouting
Roy - Carolyn
RETIRED DOAF/DON/DOD/CONTR RADIO TECH (42yrs)
K9PHT (Since 1957) 146.52M
2010 F150, 5.4,3:73 Gears,SCab
2008 Starcraft 14RT EU2000i GEN
2005 Flagstaff 8528RESS