I've in the exact same position. All of this is what I think but I'm not an electrical engineer. I checked my wiring/breaker and GFCI is most outlets and the tv outlet can be rewired from the fireplace outlet to the gfci pretty easily since both sets or romex are in the tv cavity.
You can check to see if the charger needs to be updated with lions, I did charge my lithiums individually with the existing one but pretty sure the battery mfgs recommend a lithium charger. That is an easy swap from what I see as there is a direct replacement from progressive dynamics. I was looking at the Freedom X or the Freedom XC which allows you to not need the charger. There is just enough room to mount the 1000X on the panel under the drawer and have almost the mounting clearance they recommend, it is about .5" short of the 5" on the fan end and 2" on the wiring end. But they also recommend the batteries be within 5' of the inverter. So you can put the batteries in the closet area otherwise everything needs to be in the pass through.
If using the pass through now you are running romex to and from the inverter (interrupting the gfci romex out of the breaker) through the wireway that runs from the breaker area to the shutoff area you have photos of and then rewiring the batteries a bit. I'm not sure how difficult that is but that is making the change. You also need a fuse and might want a shunt with battery monitor. Many also show a manual shutoff but i'm not sure why the existing one will not do the trick. If mounting in the pass through I would go with the 2000XC and bypass the charger. Now that is $650ish.
Now you can leave the batteries on the tongue but they won't charge below 32 degrees and I would want a locking box which is adding a few hundred bucks. So they should go in the passthrough or I'm looking at under the bed.
Then I think that instead of running the gfci back and forth why not just run one leg of the breaker back and forth and invert half the breaker which adds the microwave and fridge which will come in handy when driving and not wanting to run the fridge on gas. But now that runs about $1200.
You can check to see if the charger needs to be updated with lions, I did charge my lithiums individually with the existing one but pretty sure the battery mfgs recommend a lithium charger. That is an easy swap from what I see as there is a direct replacement from progressive dynamics. I was looking at the Freedom X or the Freedom XC which allows you to not need the charger. There is just enough room to mount the 1000X on the panel under the drawer and have almost the mounting clearance they recommend, it is about .5" short of the 5" on the fan end and 2" on the wiring end. But they also recommend the batteries be within 5' of the inverter. So you can put the batteries in the closet area otherwise everything needs to be in the pass through.
If using the pass through now you are running romex to and from the inverter (interrupting the gfci romex out of the breaker) through the wireway that runs from the breaker area to the shutoff area you have photos of and then rewiring the batteries a bit. I'm not sure how difficult that is but that is making the change. You also need a fuse and might want a shunt with battery monitor. Many also show a manual shutoff but i'm not sure why the existing one will not do the trick. If mounting in the pass through I would go with the 2000XC and bypass the charger. Now that is $650ish.
Now you can leave the batteries on the tongue but they won't charge below 32 degrees and I would want a locking box which is adding a few hundred bucks. So they should go in the passthrough or I'm looking at under the bed.
Then I think that instead of running the gfci back and forth why not just run one leg of the breaker back and forth and invert half the breaker which adds the microwave and fridge which will come in handy when driving and not wanting to run the fridge on gas. But now that runs about $1200.
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