I use this strap wrench. Works fine all the time. You already know what can happen using your #1 method.
Before putting the pulley back on, you can spray some lubricant, but hitting it a few times with a rubber mallet is fine too.
A gap of 0.9mm is way to much and you should address it immediately. The gap should be 0.25mm-0.3mm.
And don't use the air gun for the pulley. it will bite you in the rear one day.