It sounds like your string lights
are AC powered if you plan to use an inverter, and the inverter itself will use some of the battery power. Are the lights
LED? That will allow use of a battery about 1/10 the rating of lights that are incandescent.
controller wouldn't be needed for such a small wattage panel, but I really don't know about the effect on a small battery. I don't think that your charger will overcharge a smaller battery (you would top out at 1 amp per hour), but I'm not an electronic expert.
As for smaller batteries, they do have small motorcycle batteries, and also riding mower batteries that should work for your needs.
If you ever plan on getting electric trailer brakes
then perhaps a battery designed to operate the break away cable would take care of all your needs.
Or you could just go for a standard type 24 marine battery at WalMart or Sams club.