Why MOSFET transistor doesen't do the switching?
filament gets red just slightl + MOSFET starts heating up a lot

This means that the MOSFET is not fully on. The Vgs_max of the IRFP260N is 20 V so you can just feed the +12 V to the gate.

The transistor Q2 is also used in a pretty useless way. Let me propose a different schematic:







simulate this circuit &ndash; Schematic created using CircuitLab

If M1 still does not fully turn on, then try replacing R1 by 100 kohm.

You don't need Q2 from your original schematic since there is no need for current amplification. The MOSFET's gate is basically a capacitor, it just takes longer to charge with a small current. But since this is for a lightbulb, it does not matter if charging the gate capacitor takes one second. The lightbulb is just as slow.