Can you go up and around, then back down? I think a channel would also work like you mentioned.

Your idea of running it in a “trench” on the drywall is not too bad, but don’t use mud alone, use something like 1/2” resilient channel, so there is more rigidity on that joint.

And probably don’t use a router to make the trench, so you don’t go into the wood (since you are scared to drill, I don’t think you want to make a dado cut on them).

Run the cable inside the channel, butt end the drywall on it, use tape to join it, and then use mud all over that. This is not typical or proper, but it could work.

If you are not scared of redoing all the drywall, the typical use of rc channel is to separate the drywall from the wood frame, so that could also give you a gap to send cable through. (edit: this is a ton of work, I would do the trench with the channel and if it cracks go from there).