800 RPM for a M3 tap is really slow.
I would run a roll tap, and shoot for 70SFM (2266rpm) if your machine will allow.
Peck tapping is ~probably~ what is messing with your holes, or slop in the machine/holder etc.
Would a roll formed, 70% thread work? If so, shoot for a #36 or Ø2.75mm tap drill...