How do I increase 3A fixed output current of a buck converter?
You are correct that your selection of the buck converter is the limiting factor. Since you would have up to 300 watts available from the panels, ideally you would need a converter that can handle at least 25 amps for a 12 volt output. 

You may already be aware that without an MPPT (maximum power point tracking) mechanism you will typically realize less power from your panels than optimum. You should also consider that the buck converter has inefficiencies as well. An estimate of 90% efficiency is a starting point.

Then consider that your 220 volt inverter has a 90% or so efficiency. Under optimum conditions, you could power a 220 volt, 240 watt (~1 amp) load .

Do a web search for high power or high current buck converters.  The LT1339 is one example you can study. Be prepared that at this power level the design will require external FETs and that the inductor design becomes more critical (and probably not off the shelf). The overall circuit is not a simple one like your current design. Much more engineering will be needed to obtain the desired results.