It all sounds like a good idea, but the guy who built the old plywood board job with the 12v battery definitely had the right idea in the first place - use a proper 12v battery. A lightweight sealed motorcycle one should be just the ticket, and the extra weight is well-compensated by having something that works rather than lie around useless because the batteries are always flat. And you surely have the charger for it near at hand.
I'm betting you already suspect that the only 55w brake bulbs in the known universe are probably in reply #2, and are quite likely illegal, but you need to be able to handle trailers with more than one standard 6/21w anyway, not to mention heavy vehicles, so the comment about high current is not unreasonable.
I imagine this device being on an easel with the trailer sockets on a 2m lead and remotely controlled via bluetooth. I don't think you would need an Arduino at the remote end. The main unit would simply go through a programmed sequence on receipt of a bluetooth signal, i.e. the establishment of an auto-connect. The remote could be charged off the main device. The last thing you want is that to be flat.
I'm sure you could test for shorts and open circuit in the above programme.