Yes, been there and done that. Plan is to get into LAX late afternoon, air-conditioned shuttle to Lancaster and by the time we get to the storage yard the sun will be down - so by cranking the generator up and running the rooftop air cons we should be able to make it the 4 miles to the fairground where we can get onto shore power. Late at night when it is safe (and the jet lag is still raging) we go to Walmart 24/7 supercentre to stock up on food and then back to the fairground for more shore power. Next morning we hit the Interstate north and don't stop until the outside air temperature drops to 85F max. Then we will be OK for the next 5 months until we get back to Oz half way into summer.
I hate hot weather and untempered sunshine.
Back to your problem - yes, a multimeter will help a lot. With NO 12V loads switched on, just wedge the leads into the
battery terminals (so you don't have to hold them, with it on the 20VDC range and note the readings with no shore power having been connected for an hour or so. Voltage should read somewhere below say 12.9V. !2.6 up to 12.9 indicate a fully-charged
battery. Less than 12V and you are guilty of committing batterycide - or it has died a natural death.
Start the Engine. After a few seconds to a few minutes depending on how well your engine
battery is charged, the charging relay (if fitted) should kick in and your meter reading should start rising. If it does, switch off the engine and wait a few minutes before taking another reading.
Plug in shore power and turn on whatever 240V switches you normally turn on. Maybe a light will come on on your 12V control panel. Maybe not, but the voltage should start rising towards either 13.7V or over 14.2V depending on what sort of
battery charger you have.
If the voltage rises, leave the charger on for several hours and then disconnect shore power.
Battery voltage will drop rapidly (15 seconds) down to a little over 13V and then more slowly over the next couple of hours. Voltage should remain somewhere between say 12.4V and 12.8V depending on how flat it was at the start of the test.
If it drops well below 12V fairly quickly then the
battery is likely U/S.
Turn on a few lights that add up to say 4 amps and check the voltage every half hour. It should drop a fraction of a volt initially and then drop slowly - a tenth of a volt every ??? minutes - and gradually and after a couple of hours should still be above 12V.
If it drops well below 12V within an hour the
battery is likely had it, but it might be worth connecting shore power again and leaving the charger on for 24 hours to see if it recovers. If the test results are the same, then you have the job of deciding whether the batteries died naturally or whether they were helped on their way by owner neglect or a faulty charging system because there is no point replacing the batteries if something else was the primary cause.