Excellent solution @HalfNormal.
Couple of observations to keep in mind so that your project does not get delayed and/or waste mugs.
The center cut out of the foam needs to be equidistant from the edges otherwise you will get a wobble . This will cause depending on your laser lens an in and out focus.
Also keep in mind that the overall circumference for your mug on the machine is the circumference of your foam rings but the circumference on the mug is less. This means that. Mug will move slower then normal which might “squeeze” your engraving
Cheers and leave it to @HalfNormal to come up with an excellent solution