For my first kilt in DPM I used a strip of webbing that is similar, though probably heavier than your stabiliser, and that has been very sucessful.

I am trying to remember what I did with the rest of it so l can put it into more kilts.

You need to press that area of your kilt at a temperature the webbing can take, if it contains a man made fibre it will go crispy, or worse, if overheated. That means even if it is underneath other materials.

I still recall with horror the time I decided to press an errant pleat on a cotton kilt and melted a Nylon belt - luckily onto the iron, not the kilt.

Anne the Pleater