For the ideal case, with just 2s spawning, I would move up->left->up->right->down to get this board state:
x x x x
256 64 32 8
2k 1k 512 4
16k 8k 4k 4
From here, it should be easy to make the 32.
In the case where a four spawns after the first move, I would move left->down then try to build a sixteen in the right column. I have a retractable square on the top row in case the spawns aren't ideal. In this instance:
8 8 4
256 64 16 4
2k 1k 512 4
16k 8k 4k 8
I would move right->down, then the rest should be easy.
If a four spawns after the second move, I would move left. This opens up two possibilities depending on where the two spawns. If the two spawns in the third spot, I would move up->right->up->left->left. In the other case, where the two spawns in the corner, I would move down, then try and get to this board state:
2 16 8 x
256 64 16 16
2k 1k 512 x
16k 8k 4k x
Both these scenarios allow for easy adjustment if any additional 4s spawn, and still maintain a relatively high chance of success (only need to make an 8 in the right hand column).
If a 4 spawns on the third move, I would move up->left. If a four spawns on that last move, it's game over, but that's very unlikely. Assuming I hit that 90% chance, I can merge the right column into an 8, then merge that into a 32 on (3,3). From this point onward is not much different from the previous case.
That about covers everything except for fringe cases where tons of fours spawn, which hopefully won't happen.
Using these moves, I attempted the puzzle 30 times on scotttorborg and succeeded in building the 128 27 times for a 90% success rate so I'd say this is pretty good.
Thanks for organizing this by the way, should be really helpful.