Thanks guys,
I tried to put on 100nF + 100 ohm snubber, nothing changed.
That is true I can't measure the exact current, I used a multimeter.
I examined the Voltage on the motor on scope, it goes from 15 to 18V as load changes (motor runs) so the current does the same I think, so the 3A is just a raw average.
I found a STP60NF06 in the drawer which I salvaged from another screwdriver I tought I'll give it a go.
It's 60A / 240A pulsed, with Rds 0,016 Ohm.
TADA: it works like charm.
I noticed the motor runs faster instantly, it does get a bit warm after 10-20 sec but way much better.
Since it primarily used for screw in and stop like movements, it will do.
Could the Rds 0,016 vs 0,044 (IRF540) count this much?