You just need to be super-efficient.
Roaches:
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×Do them immediately after grabbing the speed potion, while you still only have one builder to contend with. I did the middle three first (anticlockwise), then the corners (lower then upper).
Rattlesnake:
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×I waited next to the lower entry until the snake went past and I could cut its tail immediately after stepping in. Try to cut every half-turn if you can; you can waste a few on turning/stepping but not many.
Mud:
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×It's possible by entering from the corner square. Just be careful not to step on either of the other exits when cutting in or it won't count.
Neither of the last two seemed to depend on having a builder in a specific spot from previously; I just waited until they were both settled at the relay station.