Yes, the refer needs 12V power for it's control circuit board.
Your water pump is also 12V. Since you don't have a hot water heater you won't be running it for showers, but you'll still use it some.
If you have room for a second battery I would certainly install one. That doubles your available power in one simple stroke. It will take a couple of extra cables, but you can get those at a NAPA. If you do this, you should use two 6V batteries as they provide more power and are meant to take longer and deeper power draws than 12V batteries. You will have to top them off with distilled water every month or so but very easy to do. On the internet search for The 12V Side Of Life it will cover wiring in the 6 V batteries - 2 of them give you 12V.
From your pictures it doesn't look like you have AC, but I'll assume you run the vent vans in warmer weather or to remove humidity when it's raining.
The LED light you bought should work in any position - think of taillights with their mounting bases in all sorts of positions. I'd double check the bases on bulbs in the dinette fixture, are they different than your other bulbs? Again, the LEDs should work in any position.
You mentioned 12V maintenance. If you get a Honda 1000 or 2000, the maintenance is really almost zero for years and years. All you have to do is change the oil which is very easy. It takes about a 1/2 qt per change every six months.
I'm assuming your TT has a built in battery charger. You need to see what its power requirements are. A Honda 1000 may not be enough to run it. When you find it in your trailer, it should say on it how many watts it needs to run, if you have questions, get the model # and come back with questions. The Honda 1000 will be lighter and take up less room than the 2000. With only a one battery system a 1000 will most likely take care of you.
Do you have a furnace or a catalytic heater? A furnace fan can be a big power draw especially when the battery is a little older.
If you can install two 6V batteries, you might get by without a generator, but you might be almost to the point of tent camping in your TT. 100 - 200W of solar on the roof would go a long ways.
As the Wiz said a 900W generator from Harbor Freight is cheap. If you find that you need / use it you can always sell it and upgrade to a Honda later which would be quieter. As for noise, set it on the other side of your tow vehicle and use an extension cord.
Bill