Greetings & Welcome!
I think you have multiple problems here. A 3kwh inverter is going to draw 300+ amps with inefficiencies.
Odds are your 4x 100amp lithium batteries are only 3.6 volts, so it takes 4 of them for a 12v system wired in series. When wired in series, it ups the voltage but not the amps. Meaning that you only have a 100 amp battery bank that you’re trying to draw 300+ amps out of. Thus your wiring gets hot because your fuse is way too big for a 100 amp system, so it isn’t blowing it as it should.
If the batteries were wired in parallel, you’d have 400 amps but only 3.6 volts. In either case, you’re screwed, not even mentioning that a 3kw inverter is going to draw that many watts from the battery regardless of the load on it.
Electricity in a camper is precious and hard to come by, solar panels aren’t magic, they’re just a very inefficient battery charger, and battery power isn’t well suited for heating, cooking, hot water, microwaves, hair dryers, refrigeration, air conditioning etc. These types of things are best powered by either some type of fuel, or with a generator or shore power.
I either heat my water via solar (set a black container full of water in the sun) or on my camp stove in a pot. If I need more, I use a fire coil on my stove or in a campfire to supply unlimited hot water just like a tankless hot water heater. I’m just using my more energy efficient camp stove instead of electricity.
Save your battery power for your laptop & phone, and get car cords for them so you can eliminate the inverter all together. If you want to run household type stuff use a generator or shore power, not an inverter.
You’re also going to want to charge your house batteries via an isolator when you’re driving, which is much more efficient and reliable than solar. I skipped solar in favor of a cheap generator & a cheap battery charger. Total cost of $129 and totally reliable. 99% of the time, simply driving keeps my house battery charged, and the other 1% of the time uses only a couple gallons of gas per year for the generator, and if I need to run something that requires household power, the generator will happily supply it.
There’s a decent chance that you’ve ruined those lithium batteries. If you have, replace them with normal lead acid deep cycle batteries from a junkyard for cheap. Don’t waste your money on new batteries, lithium batteries, or AGM batteries. My junkyard batteries typically last for 5-7 years and cost under $20 each.
A good, efficient, reliable, and comfortable conversion should cost hundreds, not thousands of dollars. The promoters are not your friends, they’re only after your money, whether or not it’s obvious. A non factory camper is a metal tent, not an RV. If you want the comfort and conveniences of an RV then your money is better spent buying a factory RV. I’ve owned many great motorhomes for under $2500, and at that price you can drive them for years and they’re still worth what you paid for them if you keep them stock and don’t do anything stupid like modifying them. Modifications lower their value dramatically. Example: If you add solar, your rig is worth thousands less. People don’t want the hassles or risks involved in solar or any other non factory modifications.
A friend added solar to his factory camper, he had a flare up while cooking, (not related to the solar), and it caught fire. His insurance company denied his claim because his camper had been owner modified. You want to keep everything as cheap as possible, easily replaceable, and never count on insurance to pay you anything. Your real insurance should be an emergency fund large enough to repair/replace everything, plus at least a years worth of living expenses. For me that’s a minimum of $10k. which includes replacing everything, and a years worth of living expenses, mine does not include the money for rent if I needed to rent somewhere, but I also have a large retirement fund if I need it.
It is much harder for someone to be a successful nomad than it is for someone with a stable job & housing.
"Always avoid expensive solutions to cheap problems." ~ OffGrid