^Sounds like you’re tracking correctly. And since you have a new relay, use that.
My comment (as I’ve done similar to several trucks/boats/campers) was in regards to the type of relay specifically. If I’m repeating what you already know, apologies, but maybe it’s helpful for others too.
There are obviously different types of relays/isolators/combiners and my understanding is as follows.
Simple relay with 12v trigger (typically ignition on signal for something like your application). Signal latches the relay and it then effectively puts the batteries in parallel for both charging and load. Like wiring them parallel with a disconnect in between.
Diode type isolator. Similar to above but the diode blocks reverse current flow. IE power can flow into that battery on the downstream end (charging or upstream battery will equalize down into the downstream while triggered to “on” if the downstream is lower voltage.
Auto sensing relays. No ignition trigger. It senses voltage above nominal full charge battery (12.6-12.8V) and when that is sensed it latches the relay to allow charging and use of both batteries seamlessly. When that voltage goes down but not low enough to kill the battery, it disconnects preventing from draining both batteries from a load that is coming from one or the other.
I prefer the latter. For a couple reasons. It doesn’t match with just the “key on”. It doesn’t require a voltage source to latch the relay. And it allows for equal use/charging/balance of start and house batteries whenever charging voltage is present and even when it’s not, down to a certain voltage (preserving adequate charge in the start battery).
If I’m off on my presumptions I’d Appreciate comments as this is just my shade tree understanding.