I've faced this issue with knee braces. How about an old-fashioned (but maybe masculinized) garter belt? You can get the metal garter parts cheaply at JoAnn's Fabric. No warranty wrecked, hopefully, and he can adjust via the garter belt adjustment (like a bra adjustment) that you can build in. A garter belt has two garters, but maybe he needs four. Still, it's a cheap solution and an afternoon's work.
If you do sew on the stocking, use a rounded point or ball point needle and, I agree, don't take the current seam out. I would not use a serger (I know you like vintage machines) because it is intended to cut the fabric. I would use a zig zag stitch if possible, one that takes smaller stitches in every zig zag, i.e., a stretch stitch.
I hope he does better every day. Hang in there!