You have to have precise machine to do many parts both side at once, successfully i mean. Most probably you will spend long time figuring how to do one part right and then try many parts at once.
Inserting things will or can lead to further difficulties.

The correct way as i see it is to Zero All, surface the bed, or some sacrificial piece fixed over the bed. Zero Z. Then cut on the now flat bed the working sheet in some rectangular form. Zero Z. Then cut the same form recessed in the bed so the previous shapes have tight fit. Then you can fit the working sheet in the recessed hole. Zero Z. Then do the one sided job, then flip and do the same at the other side. And so on.

So no second zeroing all and don't use the reset button or turn off the motors so they can keep position/assuming steppers/. It would help also if you check again the Z0 when flipping the sheet. It would help further id at the XY Zero you make the program to drill a hole, so you can see eventual mismatch before you started destroying material.

Thats how i do it on a cheap machine.