The problem with running the fridge and A/C offa an inverter is that they have pretty big draws and they run forever.
If they draw a combined five amps AC, your gonna be pulling about fifty amps out of your batteries!!
No battery bank will stand up to that very long. Inverters are fine for running heavy loads for short periods of time, or small loads for several hours if you must.
I'd wire the trailer for DC lights. I like flourecent fixtures. They are less money that LED stuff, use standard off the shelf bulbs, and throw a heck of allot of light. Also, DC flourecents do not flicker because they run at such a high frequency ( something like 4,000 Hz, as compaired to 60 Hz for regular AC flourecent fixtures ).
Something like this -
QUOTE
L- 139: 32 watt 2.4 amp, with F32T8/CW tube 2850 lumens 48":...............shipping $8......... $54
L- 116: 30 watt 2.1 amp, two F15T8/CW tubes, 1740 lumens 18": $42
L- 193: 15 watt 1.3 amp, with F15T8/CW tube, 870 lumens 18": $34
My favorite is the L-116, which I use in my cottage, but the L-93 would probably give you all th light you need.
Of course, compact DC flourecent bulbs with edison bases are availible -
QUOTE
L-CF 7: 7 watt, approximately 325 lumens, total length is 5 1/2": Warm yellow color $18
L-CF 11: 11 watt, approximately 600 lumens, total length is 6": Warm yellow color $18
L-CF 18: 18 watt, approximately 1200 lumens, total length is 6": Warm yellow color $24
L-CF 23: 23 watt, approximately 1800 lumens, total length is 6 5/8": Warm yellow color $24
L-CF 7: 7 watt, approximately 325 lumens, total length is 5 1/2": Cool white color $18
L-CF 11: 11 watt, approximately 600 lumens, total length is 6": Cool white color $18
L-CF 18: 18 watt, approximately 1200 lumens, total length is 6": Cool white color $24
L-CF 23: 23 watt, approximately 1800 lumens, total length is 6 5/8": Cool white color $24
I have a few of these, but my only problem with them is when they burn out you must mail -order a replacement. Not that I've ever had one burn out...
You can find small DC flourecent fixtures like this;

Or this;

Or this;

I'm sure you could find something that would suit your needs.
Sure they draw more than LEDS, but they give much brighter light. Trust me, I have a few LED lights and all they are really good for is reading lights and such.
I'd pull the AC fridge unless you expect to spen allot of time in a trailer park hooked up to utilities.
I'd just use an ice box myself. Maybe you could find a used RV propane fridge, but installing it with the vents, gas piping and such would be a pain.
Convert the trailers AC wiring to DC, and switch the lighting fixtures over to DC and your all set.
Buy a good sized inverter, install it close to your battery pack and use HEAVY cables to hook it to the batteries.
Make sure you can easilty switch the inverter on and off, because you'll only be using it for short periods of time. Use and extention cord for the AC output or rewire a few outlets in the trailer.
We use an inverter to run our TV when we fell like watching a movie, or to run small kitchen appliances ( No Toasters!!

), and that's about it.
You might want to look at something like this -

This 200 dollar kit from harbor freight has 45 watts of solar panels, a charge regulator and two lights!
All you'll need to add is some deep cycle batteries and a cheap inverter like this one for from Harbor Freight.