One kilowatt output should be more than enough to take down a missile.
Assuming 30% efficiency for the laser you would need a 3-kilowatt generator.
I don’t think you would need a separate engine to power an on-board laser weapon.
But unknown is how heavy the laser will be. Will you need more powerful engines to lift the extra weight or will you have to leave other ordnance on the ground to carry the defensive laser.
Supposedly the F-35 would use the lift-fan PTO to power the laser, so the F-35B wouldn’t carry the laser. The ‘A’ and ‘C’ models would be capable.