Best method to level 1x3 strapping to ceiling joists on a slope?
This is less hard than you think. Last one I did was ~10x20, and it took about 4 hours. A laser is extremely helpful for this task, though I suppose you could do it with string and a bubble stick. (It is, however, too early in the morning for me to think my way through all that, so my instructions have frikkin' laser beams.)

Cross-strap the ceiling 16" OC with one small, smooth (like, 15g, 2" is good -- no spiral or ring shank) nail per joist intersection. Plan ahead for sheets of drywall.

Find your lowest spot. This is where everything is coming down to. Make a story stick (a couple of inches shorter than your low spot) and push it up to the low spot. Mark the laser location on the stick. (Mark the stick with "this way up" if you're having one of those days.)

Now, for every intersection, tap the strapping down so that your story stick mark matches the laser. Shim as you go, but don't put final nails/screws through yet.

Once the whole room is done, doublecheck with the story stick. You can be as precise as you want, but if you had, say, cabinetry and crown, then you'd want to be super accurate in those areas.

Snug the shims and nail or screw everything off. If you don't nail through the shims, you can make micro-adjustments afterwards. I've found that 2 nails per junction of 2-1/2" non-smooth sheathing nails are perfectly adequate to hold up the strapping and rock. Doublecheck if you want.

Slight ps, I use 1x4 to give a bit more wiggle room and allow for not-exactly-straight-stock. The upcharge is nominal.

Double ps, this assumes a ceiling only slightly out of level -- up to 1". On second read, I'm thinking there's a possibility that you're dealing with a very sloped ceiling that you're trying to flatten. ("Level" might have confused me, as it's a term with very specific meaning in my world.) If so, do all the cross-strapping and light nailing, then use a web of string to bring all the junctions to the right point.