You can certainly do what you want to accomplish, but you can’t charge your battery using the unit you linked to, which is an inverter for converting a battery output (DC) to mains. To charge a lead acid battery you want to go in the other direction from mains to DC and a normal car start battery charger is not likely to give you enough charging current to refill a 130Ah battery during 6 hours of peak sunlight, so you’d want something quite beefy
like this one or even bigger.
Then to run the PC at night your inverter you linked to might work, but it’s very expensive, there are cheaper ones which are still good.
But… it’s inefficient (and expensive) to convert from battery voltage up to mains and then convert back within the PC PSU to DC which the circuit boards need. Would you be willing to open up the PC and replace the PSU with a DC input version
such as this one? Then you can run the computer direct from the battery and you still have the fallback option of using a bigger capacity mains to 12V PSU to run the whole setup from mains if your battery happens to be empty.
If you try to do it with an inverter, don’t underestimate the size of inverter and battery you need, as the surge current when the PC first switches on might cause a 1kW inverter to shut down. A lot of the Chinese made inverters are good for only half their rated power, so think of a 2kW inverter as a 1kW unit, and devices like PCs (and fridges and freezers) can have initial surge currents when switching on which make them appear to the inverter to be a much bigger load than they are while running.