We've sort of been all around the farm on this.
1. Your first post said that you didn't think the inverter was working and you replaced the batteries.
The next step (probably should have been the first step
) is to check your Inverter for the circuit breakers. While you are at it, make sure it is turned on AND leave it on.
It's not clear how you determined that the Inverter is not working. If it's because your starting batteries go dead see my note below.
You can check Inverter function by disconnecting the coach from shoreline and with NO generator running, find the in-side Inverter control panel. It could be near your door, a panel in the galley, or a panel in the bedroom.
Turn the inverter on at that panel (your actual inverter in the bay is/should be ALWAYS on.
Now turn on some 120VAC light, your TV, or the fridge. Don't try the microwave or Air Conditioning - the Inverter can't run them.
You now know whether your Inverter is working or not.
If its not working, either you have an open circuit breaker, a bad connection somewhere, or you haven't turned it on. If you can't sort it out at this point you will need professional assistance.
2. With the Freedom 10 the question about transformers is irrelevant. Forget about transformers. Your system is much more sophisticated.
3. If the Freedom 10 is working properly than you do not need the auxiliary charger.
Note: If your coach is wired like the majority the shoreline power will NOT charge the chassis batteries. You need something to carry charging current from the house side to the chassis side. I have the Trik-L-Start , a $50 device that can be permanently installed. It keeps the chassis batteries charged while the house batteries are getting charged by the convert built into the Freedom 10.
Ultra TRIK-L-START Starting Battery Charger/Maintainer
Finally, you will be using the inverter to produce 120VAC when you need it and have no other source of 120 (no shoreline and generator not running).
I run mine while driving because I have a laptop at the dash running, and when I installed a new rear camera system I got lazy didn't run a 12VDC to the monitor - I just use little 120 to 12V power cube.
The Freedom 10 will keep your batteries charged (see note above) when ever you are on shoreline or are sitting with the engine off and the genset running. When you are driving the coach alternator will keep both sets of batteries charged.
Hope this summarizes all of the responses and helps you understand your coach electrics a little better.