It's done... I hemmed the inverted pleat practically the same way as it had already been done by HoE on the left hand side of the apron. It's now nicely tucked away under the apron.

I also tried holding the inverted pleat closed as suggested. Unfortunately that then pulled the first normal pleat open so I didn't stitch it closed.

Oh and as for the "successful tackling" I have to confess that for the underapron strap I stitched through the whole kilt. Had I tried to sew it on without going through the lining, there would have been only one layer of fabric to hold the whole weight. Would that have been right?
Are there any reasons not to stitch through the lining (besides aesthetics, of course)?