No need to disconnect any cables. Just plug into shore power without the trailer plugged into tow vehicle and measure the voltage across your terminals. If it reads >13V, then the converter is working. 13.6V is the voltage your converter will pull the batteries to, meaning it is doing what it is supposed to do. There is a chance it could read 14.4V if your converter is really working well, but if it is a WFCO brand then it will almost certainly read 13.6V. A functioning converter will have no problem holding steady at 13.6V with your battery bank, despite whatever 12V loads you may have operating at the same time.
$250 to 500 sounds a bit high for a replacement converter, unless that quote includes labor.
This website is often suggested for reputable advice and pricing on a replacement converter. Talk to Randy is what "they" say
BestConverter - Converters, Inverters, Electrical Supplies, Electronics
If you will be boondocking and needing to recharge via generator, then you would be looking for a replacement converter that reliably starts charging in bulk mode for getting the best charge you can in the shortest amount of time. This means the converter tries for a 14.4V or higher setpoint while in this charging mode. The OEM WFCO converters are supposed to do this, but typically don't in practice which is why I say they aren't worth investing in if you'll be using a generator for recharging. Despite me talking bad about them, I've never replaced my WFCO. If I did, I would probably get a Progressing Dynamics or a Boondocker.
If you will not be relying on a generator for recharge while boondocking, then this bulk mode scenario won't really matter to you.