First, there is no "charging circuit" in the Giggy Box. All that is in there is wiring connectors, 12 volt DC mini-breakers and an off/on switch. All the "charging circuitry" is in either the solar system components and in the WFCO converter/charger.
As for whether "YOUR" converter/charger is fully compatible, partly compatible or not compatible with lithium battery charging configurations is only going to be found by knowing "EXACTLY" which WFCO converter/charger is installed in your trailer and by going to the WFCO website to either research that converter/charger "by part number/model number" or by contacting WFCO tech support with that information and asking them for answers to your specific questions.
Here's the link to the WFCO website:
https://wfcoelectronics.com/
On that site is a section called: "LITHIUM SOLUTIONS"
https://wfcoelectronics.com/pastgeneration17/
Many of your questions would be better answered by the WFCO engineers rather than by "armchair experts on a forum"....
That said, once you get the WFCO answers and want to know if anyone has used the converter/charger that is in YOUR trailer, then you can get the personal experience of users of your specific converter/charger.
Your "open ended question" "will my trailer support lithium batteries" will get you everything from "WFCO builds junk" to "Buy a Progressive converter/charger and you'll be fine" to "my solar system keeps mine charged, I don't even know where my converter/charger is located"....
I'm not "passing the buck" by not answering your question, rather suggesting that without knowing WHICH converter/charger you have and what kind of system you want to build, it's impossible to say that your converter/charger will or won't work.