I thought the bottonholes were made out of the yarn end after you bind off (so type of CO wouldn't matter anyway)? That way you have as much as you need to make the loop.
If not, you can just chain (crochet) a loop on each end of the strap.
If you picked up a stitch on the end of the strap edge, can you do a knitted or cable CO, then just join it at the other edge?
"The cat has too much spirit to have no heart." -- Ernest Menaul