Worked it out.
×The key is that there are four positions on either side you can push Beethro diagonally from, so you can just use all of those to minimize conflicts. All the sticks wind up on just two tiles, so you just have to make sure that the right timeclones move first. (Do this by recording the ones that can potentially get blocked first, so they wind up being later in the turn order.)
Full spoiler if the above explanation wasn't clear enough:
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9s are where the sticks go. The rest of the numbers are recording order.
All northeast clones move southwest, all southwest clones move northeast. Make sure no clone is delayed so they all push at once, and make sure you leave enough time for the main Beethro to get a sword and get in place.