With the ability to use the Flying Spaghetti Monster's light bridge, the Pastafarian can easily weave between solid tiles and bridge tiles to avoid all the boxes, and potentially clear the level without needing to jump at all. The Lizard can do long, high jumps, which is very helpful for minimising the number of jumps used, and furthermore his slow speed can make it far easier to react to the boxes in the level.
The Bunny may be a good character, because bounces may not count as jumps, although this is unconfirmed.
The Gentleman actually works really well because he spends most of his time in the air. You can use the Gentleman height trick to stay in the air. Just be careful not to slam right into a box. You can avoid this too by pressing left or right while approaching a power cell.