The battery temperature sensor can be on any battery, as it has no electrical connection to the batteries. There is one other cable that needs to be relocated from the battery on the left, move the TM 2030 Battery monitor to the battery on the right, so it can monitor all voltage/current properly.
As an example, the current flowing through the first battery can be different that the current flowing through the second battery. In other words if your system was getting a 10 amp charge from SOLAR or TT Charge controller, the battery on the right may be pulling in 6 amps leaving 4 amps to the battery on the left. Your monitor may not display the correct numbers. This is a little extreme example, but you get the point. This is especially true if the batteries are not the same age, mfr or Ah rating.
On the right side of your Inverter picture, the voltage is 110VAC, remove the (+) and (-) from your drawing.
You will also need to install a new 110VAC breaker in your main AC panel (mine had 2 blanks for future use) and wire your TT's DC charge controller to it. When you are running without shore power, turn off the breaker as you do not want the TT's battery charge controller to be active (trying to charge the batteries with the batteries). You could wire in a relay with the Automatic Transfer switch to cut power to the TT's battery charge controller. We also have a picture of the thermostat with a RED line through it, hanging over the thermostat.
Also, to avoid any possible accidents, turn off the Air Conditioner 110VAC breaker when not on Shore power.
I am not sure as to what Ah batteries you will be purchasing, but for example the Trojan T105 (6) volt batteries are around 220Ah, of which you can only use 120 Ah. Yes, the bigger the Inverter output rating the higher the idle power it consumes. If you are only going to have (2) 6 volt batteries (around 220Ah) I would just get a 1000 or 1500 watt inverter, they all have a surge limit built into their design.
Hope this helps a little.
Very NICE schematic!!!
Don
My Registry
RVing with SOLAR