I'd be apt to do some form of Russian Join, especially given how narrow the strap and the stress factor. And I'd probably extend the join, working with both strands over several sts both approaching and leaving the actual join. (Not familiar with the yarn if working with extra strands would be really noticeable.) I'd finish with dupe st of any tail length and avoid making the join ends too short. (Just in case you need to re-secure them at some point. If you lock the ends with FrayCheck that'll help with them potentially poking out.)
Not sure how much that helped.