didn't read the whole thread but here is my take:
you have a gaming computer, which draw's lots of current. I have a feeling you live in an area where you have frequent power shortages. You must have put your batteries to work during all those power outages. 2 batteries were enough in the beginning to juice up your high current system but now the time has come the 2 batteries can't charge up to max after 2 years of usage. The plates inside the batteries have corrosion and charge can't be built up to the max like when they were new. Maybe you have never bothered to check the water levels in the batteries all this time. when the water level goes down the batteries begin to go bad quickly. Always use battery water purchased from the store.
It seems several of you are confusing two related but different concepts: energy and power. Power (measured in Watts or a multiple like kilowatt, megawatt, etc) is energy per second, that is the rate at which you consume energy.
A simple example with water: a reservoir holds a certain volume of water, measured, for example in gallons. The rate at which you consume water from the reservoir (gallons per second) is the equivalent of Wattage in electricity.
The maximal Wattage that an inverter supports tells you how fast you can consume electrical energy, that is how many Joules in a second. But the total amount of energy available to you is determined by the batteries (this is the reservoir of electrical energy).
The overload is caused when too much power, that is energy PER SECOND, is drawn. This has nothing to do with the batteries. You could have dozens of fully-loaded batteries available to you, and STILL have an overload situation if more energy-PER-SECOND is drawn than the inverter can support.
Likewise, with a very powerful inverter you may never have an overload. BUT if too few batteries are available (the reservoir is too small) the amount of available energy will run out quickly. Thus everything will work perfectly, but for a relatively short time, after which the batteries will be dead and you will need to wait until the batteries are recharged.
So, changing batteries cannot be the solution to an overload.
If the OP doesn't want to purchase a more powerful inverter (and he says his blackouts are relatively short and infrequent) a workaround is simply to reduce load while on the inverter...replacing standard light bulbs with low-energy ones is an obvious measure (if not already done) which may also save money in the long run...so-called compact fluorescent lighting use about one quarter of power for given light output (the consumption from 5 100W incandescent bulbs can be reduced to slightly over 100W relieving the inverter from almost 400W just like that...unplugging the fridge should work for a few hours.