Hi Ricky welcome!
The external transformer then is connected across the 240v and feeds back on an isolated.Winding the 110v the pj control needs, to be in balance. This mod does cause the pj control to drive primaries 1 volt higher than original design. In some cases with low battery voltage it could cause a problem.
At one time I had what appears to be the same connection setup you are describing.
On my setup, the control board was monitoring 110v to be in balance but that was true only because it was originally a 110v/24v 60hz cntrl board. I had added the 240v split phase part.
What I didnt realize at the time I wrote some posts in this thread is that the cntrl boards designed for 110/220 normally monitor the 220v not 110 as you describe. I'm curious, was your control board originally a split phase board? Or did you change it into split?
You are absolutely right on about the battery voltage. Thats why i eventually changed the AC volt increase method to a few turns on each toroid secondary and series connected each to AC output without letting cntrl board monitor the boost.
For me low bat volts was never an issue though since i used lithium 24v at that time.
For those that are concerned about keeping AC voltage up to 120 there are two other factors that should be mentioned.
1) wire sizes, if you have a fair distance to transport your AC power you will not regret putting in heavy wire. In USA120/240 split phase service to a residence is often run with 4/0 (107mmsq copper even for distances of 100feet or 35meters. That may be an overkill for an inverter but some guys are reaching incredible power levels and it takes wire to do it without v drop. I choose AWG 6 (13mmsq) and so far it has been solid at 30amps/phase 7kw. My fuses blow after that so its sufficient for me but everyones needs are different.
2.)Unlike most everywhere else, in USA we use much of the power at 120v rather than 240 so efficiency can go down due to vdrop unless we balance the two sides equally. I found that out even with awg 6. All i guess im saying is when you actually watch the numbers during motor starts its more important than a guy thinks.