Well, if you take a close look at the following screencaps:
You can see that the PADD has a significant thickness to it. As such, I think the the side fin plant-ons can’t possibly be only 1/8" thick.
I was thinking just under 1/2" thickness for the side fin plant-ons, with the main body between 1/2" to 3/4" thick. Definitely thicker than Geordi’s S7 PADD.