Although for now I've found a solution but still wanna keep updating here as I get more knowledge on the issue.
There's something called "jerk load" (sounds so wrong lol). This is the power/current drawn by appliances when there's a sudden interruption in power (like during switching from mains to battery). These appliances will try to stay on by drawing more power than usual during interruption. For the inverter to support this scenario it must be rated at least 25-30% more than your actual power needs. If the inverter isn't rated that much then it'll trip and show overload condition.
I'm guessing this is what is happening to me in addition to the switching time not being enough for my PSU. Switching time needs to be less than 15ms for stress load but inverter takes 20ms. So even if I had a higher capacity inverter I would be in the same situation of my PC getting restarted when on load and power cuts, only benefit that inverter won't show overload.
I might get higher capacity inverter later this year or next year. Not really needed now.
