The inverter is not really determined by the size of the batteries, but rather the size of your max demand. In a battery system you would generally work out what the max amount of AC loads will ever be on at one time and add that up to be sure you have enough inverter capacity. IN other words, your loads determine your inverter size Is this a bimodal system where you will also be selling excess solar generation back to the grid? If so then it would also need to be large enough to handle that. The other thing to keep in mind is that you should size your batteries enough to handle your loads. Two 12V 100Ah batteries is VERY small. This would be about teh size of battery bank you would have to run some LED lights, charge a cell phone, that sort of thing. Thats basically two car batteries so think more along the lines of what a car would be able to power. If you tried to put a 3kW load on that small of battery the voltage would sag hard and instantly hit the LBCO. Even if you eased into it it kept the system on a 3kW load will kill the batteries in less than an hour and you would hit a typical max discharge of 50% in like 15-20 mins. On that size of battery you would probably want to try and stay under 200ish watt AC demand. The first question to start with will be what are you trying to do with this sytem? What would be the peak AC demand you will ever need, and how many kWh are you looking to use each day. With this info you can then determine everything else More on reddit.com

A 2kW inverter will draw about 2200W when outputting 2kW. The battery BMS must be able to supply at least 200A continuously. If it's not rated for this, then it will shut down. The cells might tolerate it, it would be close to 2C rating however would be better to watch how warm those are. At such power ratings, every connection has to handle the current, which is quite huge. A 100ish fuse has a delayed time based on current. At 200A it might list only 20-30 seconds, which is fine for the battery but I guess not fine for your load. More on reddit.com

You seem to be burning through a lot of power. I rarely use much more than 1200WH (Equivalent to your 100AH) in a day and I don't think about my energy consumption at all. Run my computer, fridge, and pretty much anything I want for that much power. What things are constantly on for you? Are you running an inverter full time? Do you have shunt connected to your system monitoring your usage? More on reddit.com

I did exactly what you are proposing. Started with a 3000 watt inverter. Then months later upgrade to Lithium batteries. Six months later I added solar an DC-DC charge controller. We traveled and the system worked great at all steps. For me the 3000 has been overkill, AC is only appliance uses enough to justify and in 45 min batteries are drained. Oversize doesn't hurt anything but budget. Regardless of size of inverter, it will only draw equal power from battery as you are consuming. A 10 amp (1200 watt) load will put the same load on your battery from 1500 inverter or 3000 inverter. More on reddit.com