Room lights flickering when I connect my desktop computer when running on Inverter

Your desktop computer's PSU draws electricity in a more complicated way than something like a light or fan does. Nothing's really working incorrectly... it's just that the inverter and the power supply weren't built for each other. The power supply is sucking power when the inverter doesn't expect it, causing a dip in the voltage; then the inverter up-regulates the voltage to compensate, only now the PSU isn't drawing any power and the voltage goes too high.

Replacing the PSU with one that does "power factor correction" may fix things. (Be careful - manufacturers of cheap PSUs will flat-out lie about whether a PSU does power factor correction, as they lie about other specs. Buy a trusted brand.) Alternatively you can use a voltage regulator, such as this ugly little box, to isolate the weirdnesses of the PSU and the inverter from each other. A good voltage regulator like that one will instantly solve a wide range of power issues. (I swear by the APC Line-R, but other people prefer the Tripp-Lite brand.)

Solar or other off-grid power is 10% generation, 80% conservation... And 10% selecting products.

The first thing that concerns me is plain overload. The problem is "every light and fan in the house" is not a scientific unit of measure. I really wish we had something like a Kill-A-Watt to measure each light and fan. We are concerned with both watts and VA. Watts is the part of the sinewave the device actually uses, but VA is the whole sinewave which the power source must generate. The ratio is called "Power Factor".

Bad power factor is how a low wattage device can overload a large inverter.

A big LG tower claims to be 250 watts at 85% in its marketing material, so that would be 285 VA. (I would be happier with the actual numbers on the nameplate). And that is worst-case (max load)/best case (highest power factor at max load), the PF could be notably worse at lower loads. If the PC has been upgraded with an aftermarket power supply, all bets are off - an "850W" power supply actually wants 1200VA.

A competent quality 150AH battery + 1000W inverter combo goes for about $450 in the US market, so your $360ish is sensible given lower Indian distribution costs. Of course, this means nothing at all; the latest trick of the junk sellers is to set prices near quality units. Another plus is that they have a customer support line and do try assembly level repair on the units. However they seem wobbly on the actual capacity of this thing. The "1550" number suggests 1550W yet their own marketing material suggests 950 watts or so. Maybe it's 1550W peak.

I would be happier if you could get a power monitor like a Kill-a-Watt and survey each of your loads. Better would be to put it on the inverter output, and closely watch both VA and watts as you run different loads. 250 watts is a lot of power by comparison to your other devices, so I suspect the PC may be overloading the inverter when combined with the other house loads.

One other thing: don't put a UPS on a UPS. If you are running this battery system to deal with short term utility power failures, then plug the PC's UPS into the mains power, not into the Eco Volt. Let the PC UPS back up the PC, and the Eco Volt back up he rest of the house.

If you are entirely off grid on solar etc., then get rid of the UPS since the Eco-Volt already does that... then remember what I said about solar being 80% conservation. Your PC is too inefficient... select a more efficient one. Mac Mini's from just a few years ago are high water marks for cramming an impressive amount of performance in quite a small energy budget. And their power supplies are quite good on Power Factor. There are many other micro-PCs, and these might be better still if you can get a power supply that runs on 12V directly (no such exists from Apple). However the micro-PCs I have seen are weak tea from a performance PoV and their performance:energy use ratio doesn't match Apple. You can run Mac hardware Windows-only, or Linux only... no need for MacOS.