hopefully they will resolve your issues at a reasonable cost... :-)
The summary that you have provided is very good. If you would have tried checking things with a meter (mostly simple checks), odds are that you would have found the issue. Having said that, based on the summary, it appears that the "resetable breaker" in the main line going to the battery needs to be either replaced or reset. It is possible once that is done, your converter may then see an "appropriate load" and "clean up" the resulting 12v output such that your "lights observation" (pulsing) will be reduced / gone. I can't say that my weco converter (same model I think) shows this symptom but I have never run it without a battery... and if I did, I don't think I would have look close enough at the resulting lights. Hopefully, that is all that happened.
As for the radio, you might be "screwed" a bit. Your "reversal" would have been no different that doing the same on your TV. Odds are high that you would have taken out some / all electronics that would have seen the "difference" even for a bit. Saving grace on a vehicle might be the ignition would normally be off which stop most but not everything.
On your trailer, it is likely that your radio was in "standby" which has lighting and possibly memory hold... meaning those circuits would have been live for the "unplanned reversal"... :-( Of course, it can be fixed but electronics like those these days are more throw away / replace as the labor charges will likely be in the $50-100 even for simple dismantle / check. A diy techy guy could probably track down the issue internally and resolve it though.