Not sure how much work you want in reframing but it seems to me, you could shrink the soffit's room depth some, and use the vertical space to host the air condition line.
Is that strapping thin? You might remove it and put up some plywood and then use Green Glue, with two layers of 5/8" drywall on that ceiling there. You'd likely end up with a
something approaching what's already there in height, but it would be more substantial.
There are almost always inches to be had, but it comes down to how much time, energy, and money you want to invest in those inches to be had?