If your camper is a newer model, it probably has a "smart" charging converter already and you could theoretically leave it plugged in continuously and it will drop to float once the batteries are full and not boil them out. Leaving the trailer unplugged without a battery disconnect engaged means the trailer's parasitic draws (propane detector, stereo, various control boards on 12V appliances) drain the batteries over the course of a couple days. This is what killed your old battery. The disconnect will shut off those parasitic draws.
If 500 watt inverter is its continuous rating, it probably has a 1000 watt surge rating. 1000 watt surge may briefly pull in the neighborhood of 100 amps which would pop your 50A breaker. If you don't ever pull that much, then no problem. I'm just pointing out that your 50A breaker may bottleneck your inverter's potential performance.
You don't mention if you used more of your 2 gauge wire to supply power to your inverter, but I would suggest doing that.
Congratulations on getting a small solar setup. 45 watts should be able to maintain those batteries while in storage, provided that they were fully charged first and you engage your disconnect. You'll need more solar wattage if you want to use the sun to bring discharged batteries back up to full.
Beware the path you're on, you now have an inverter wired up and solar input connections. Soon you might be surfing
solarblvd.com and other such sites for upgrading your off-grid potential!