Battery on trailer breakaway switch

If you want to keep the separate battery, something like the LSL Trik-L-Start would be just the ticket to do what you want. It's basically a simple DC powered battery charger, and will charge the little battery whenever the converter is running (or solar is charging, if you have solar) and the main house 12V system is over something around 13V. These are most commonly used on motorhomes to keep the chassis battery charged, but it would work just as well on a trailer if you had a separate battery for the breakaway braking system.

(But I also would vote for just having the breakaway system connected to the house battery and getting rid of the little battery as the best solution.)