I had a similar problem in the TC but with a different set of circumstances. What is the same is that the single (SiO2 instead of Li in my case) 100AH battery was good for running the inverter, but not enough AH to run the rest of the 12v normal RV loads for a couple of days as well.
I split the two jobs into two separate battery banks, which solved the problem. You have to decide which bank to use for which job depending on ease of wiring and what the batteries can do.
To solve your problem IMO, move the Li batt inside and also get a couple of Wet batts for the outside compartment--whatever will fit. A pair of 6s would be ideal. Whatever gets you the most AHs that will fit and that you can get at to check with an hydrometer and do all that usual battery care stuff.
You then can run the rig including the furnace off the Wets in the outside compartment and not have any serious wiring to do. Use the Li inside as your inverter's battery, move the inverter if need be to get the wiring right.
You might have to do something with the 120v side of the inverter if you move it, but a 120v extension cord can be as long as needed without a voltage drop problem.
Now you just have to figure out how to charge the two different battery banks. I used the OEM converter on one and a portable charger on the other. I only had truck to camper alternator charging on one bank and the solar was on one bank too. Your genset can do both at once by running the converter on one and the portable on the other though.
