I didn't remember this room so I went to check my demo. I should note one point of clarification - I moved the timeclone one south of the mark in my image, then pushed him north to that point.
As for pushing, here's a general tip:
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×An important thing to realize about timeclones is, a new timeclone will be earlier in the turn order than timeclones that existed at the time of recording. This is because Beethro moves first during recording, so the timeclone maintains that preference to keep things consistent during playback.
So, when multi-pushing, you need to figure out the pushes you're going to make, then record them in reverse. The last push is recorded first, and so on. Also, if you want a timeclone to move and be pushed in the same turn, it needs to be recorded after the timeclone that does the pushing; otherwise, it would be stunned after being pushed and wouldn't make a move until the next turn.
More specifically for this room, what that means is:
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×-At a minimum, you need to push three times in one turn to get a timeclone to the door.
-Since west/northwest/northwest works, it's easiest if you push the first northwest-pusher to the northwest again.
-However, to do this with only one timeclone would require the caber to be on the same tile as the timeclone it's going to push. This isn't possible because the caber kills anything it shares a tile with, so you need an additional timeclone to set this up - one timeclone moves into position, and the next pushes northwest.
So the order of recording is:
-set up timeclone that goes on the door
-northwest push
-move to the tile that the previous push is going to target
-northwest push targeting a tile one west of the first timeclone
-northwest push moving the first timeclone one tile west
This will resolve as:
-first timeclone is pushed west
-first timeclone is pushed northwest
-timeclone moves so that its caber will be southeast of the timeclone that just moved
-timeclone pushes that timeclone northwest, triggering a push-chain that sends the first timeclone northwest again