The radio and lighter plugs are on 12 volt
circuits. Have you checked the fuses in the power supply near the back RS corner? The door pops down and there are fuses for the 12V
DC and breakers for the 110V AC.
If your power supply is working properly, plugging in the trailer to the external AC line should keep the battery charged... it is a multi-function panel. If you want to keep using a battery tender, you can get an adapter that hooks to the battery with a plug in, then use a matching connector to hook to your charger. Motorcycle battery tenders usually come with these.
The tail lights are also 12V
DC... if it isn't a fuse, you can check the three pins that bridge the trailer to door connection. Make sure they are making contact. It's unlikely the whole LED light blew... so a fuse or connection would be the place to start. Make sure your trailer connection isn't the problem as well...
It's still not a bad idea to take the battery out over winter... but for a week or two, that's a lot of work...