This is a pretty easy fix, although I agree that a new kilt shouldn't develop popped stitches even if you wear it a lot. A well-made kilt should have so much interior construction (stabilizer, steeking, canvas) that the pleat stitching actually shouldn't be under stress even if you pull the straps really tight.
And, where is the stitching popped? At the top of bottom of the fell?
Well, on the fix. Get some 100% polyester thread (not Dual Duty or cotton) in a color that matches the tartan at the edge of the pleat. Thread a needle with a single thread (not doubled), knot the thread, and feed the needle inside the pleat from the bottom of the fell to hide the knot. Bring the needle to the offending edge of the pleat about an inch below where the stitches are popped and do a very closely-spaced blind stitch along the edge to at least an inch beyond the popped section. Go through the tartan but not the canvas, if you can feel the difference with your needle. See pic below.
Bookmarks