You're talking about a 3k budget, but that's pretty useless since we don't know the currency...
Anyhow, I'd look at a UPS in the 1000VA~1200VA range. Should last long enough with the PC and router plugged in to allow you to finish the game you're playing should the power go out, with safety time and everything in there as well.
on a few ups. you can swap out the battery for a car battery
id recomend a sealed battery and preferable a deepcycle battery
make sure to put it in a container that could take the acid if it leads, a good idea would be to tie,glue or secure in someway that it cant be knocked over.
that will give you a good long time on the cheap.
as long as you get a good unit that has a good transformer that can up the voltage to either 110/230 depending on where you live