Very clever! It's a complicated system but your tutorial and the GUI really communicate it very clearly! Would love to see a few more levels. Also loved the dialogue at the bottom.
I can see why you don't allow swapping modes - you could almost certainly brute-force some of the puzzles. However, it does feel like the "whole" puzzle is during the swap part. Maybe you could give the player N-number of mode swaps? So, move tiles, move player, and move tiles one more time? Sounds tricky to design...
Bravo! Again, nice work communicating a difficult subject!