I did try to press the motor back down and it decided to just lift up again. I didn't want to look at using tape again, it just conjures up visions of a 'live' motor shifting around again at a later date. When it lifts up the flywheel is rubbing on one of the components on the board, so I can't run it like this.
I have thought like you about making a metal strap and drilling a couple of holes and using a nut and bolt but I was waiting to see if anyone else had done something similar or have another solution.