first off hit up /r/solar for help too.
LIFEPO4 (Lithium iron phosphate) batteries are LIGHTER have a deeper depth (e.g. discharge to 20% instead of 50% which means 30% more USEABLE AMP HOURS) BETTER in colder weather (be be stored in colder than lead acid.) etc
Soft start kit / hard start kit for you AC (air conditioner) (soft start and hard start mean the same thing here, like fat chance and slim chance both mean no chance.) Google "Hard start relay kit" and end up with something like this product : https://www.amazon.com/SPP6-Capacitor-Increase-Starting-Torque/dp/B0002YTLFE for $13 that will really help with the hard starts on your inverter. Make sure you size it correctly for the sized motor you have.
You want an electrical switch that allows EITHER the inverter to power the 120 system OR the Grid tie in / generator tie in. Never both at the same time.
Likely you want a 48 volt or 60 volt battery bank instead of 12 volt. You can do a DC-DC converter to power your RV 12 volt lights and systems.
If the condenser fan works but the compressor is just humming and not kicking over then it can be a few things; the cap is either weak, your incoming voltage may be low due to demand in the area, or the windings are tight and you need a boost kit.
New capacitors are mostly made in china and kinda shitty and dont forever like the old ones, but I doubt its bad in less than a year. If it looks like its bulging and leaking oil its shot. It's probably fine and your equipment is getting old and you need a booster kit. They're cheap and easy to install. http://www.amazon.com/SUPCO-SPP6-Start-Relay-Capacitor/dp/B0002YTLFE/ref=sr_1_cc_2?s=aps&ie=UTF8&qid=1402399602&sr=1-2-catcorr&keywords=super+boost+kit
When you connect it to the capacitor you just have to make sure youre wiring it to the compressor not the fan. The current 55/5 cap you have should have 3 connection points 5 C 55. The two boost wires go to C and 55.
Just so you know there are $12 solutions for this called "hard start kits" $9-$15 range from everything from college dorm size fridges to home AC (central ac) Units.
https://www.amazon.com/SPP6-Capacitor-Increase-Starting-Torque/dp/B0002YTLFE
My meter maxes out at 10 amps, unfortunately, and RLA for this unit should be 23.4. Then again maybe this will finally be enough justification to buy a better meter.
I have this start kit on the way (https://www.amazon.com/dp/B0002YTLFE), hopefully this will give it the boost it needs. If this does end up fixing the issue, though, I feel a bit confused. Wouldn't a kickstart kit only help with the initial spool up? My unit is shutting off after ~30 minutes of use at this point.
$13 on Amazon
Takes 5 mins to install.
Supco SPP6 hard start kit - https://www.amazon.com/SPP6-Capacitor-Increase-Starting-Torque/dp/B0002YTLFE $8
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You're getting ripped off. And you really don't need it. They are only good if you're trying to gimp an old unit along another year or two and its having issues getting the compressor started with the stock capacitor.