I'm not meaning to be rude but what is the point of this?
What exactly are you trying to do?
This isn't one of the things that you would normally do in isolation; it is typically part of a larger project. It is easier for us to provide help if we can see the bigger picture.
That would require doubling the current. It's unlikely the batteries, the current switching FETs or the transformer would let you do this. Especially the transformer. They are built to handle a certain amount ot power. They cannot be made to increase that capability.
Just use a transformer with a turns ration of 2 to 1 with the device plugged into the 2 turn side. Make sure you call the firemen and paramedics before you do this. Hopefully the inverter will just blow a fuse or not power up.
A 1kW inverter at 12V isn't really very practical.You can buy them but it't generally better to up the voltage to something sensible like 48V for such high powerlevels.
All you need to do is use a more powerful transformer and MOSFETs on an existing 24V design, that's all there is to it. Use a 1kW transformer and any MOSFETs rated to 55A will do the job.
The 500W inverter that I posted all over the web uses a 12V battery. It can have an output of 1kW and use a 24V battery if you use a 12V regulator for its CD4047 oscillator and use a 48V center-tapped (24V-0V-24V) custom-made 1200W transformer.
It has a simple square-wave output that many electronic products don't work with. What will you power with it?
Also not that the appliances you've mentioned don't require more than 100W unless you've got all of them running at once and even then they still won't take 1kW.
Whay can't you buy a 500W inverter that's rated for 1kW peak?
That would be a far more sensibel option as you'll have a unit that's tried and tested and it'll be able to handle the peak demand.
This inverter circuit is garbage. It should be burned.
Its capacitors are backwards so they blow up, its transistors have avalanche breakdown of their reverse-biased emitter-base junctions (which creates a very high current in the backwards capacitors) and the transistors don't have anywhere near enough base current for any power output. It gives an output of about 25W at a reduced voltage, just before the capacitors blow up: