The general strategy is to keep the number of monsters under control. So each time roaches are spawn, try to kill the queen, or get rid of the roaches.
So: don't let a monster enter the niches without hot tiles. Only do this if you actually need this niche for your path towards the top plate.
To get you started: here's my beginning sequence
Click here to view the secret text
×Wait for a stack of three roaches in the first niche, just W of the hot tile, then move right.
Click here to view the secret text
×You now have one roach in the second niche at the bottom. Step on a swap token to make it go N and E. Three queen at the W will automatically die on the first hot tile.
Click here to view the secret text
×Make sure to swap at the right time: the queen going N and E should spawn a couple of roaches before she hits the hot tile. Meanwhile, move right to capture the newly-spawned roaches at a convenient spot further on.
Repeat steps like this until you're done
Hope this helps a bit. Don't have a demo available right here and now, but could send one over if needed.
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Autocorrect is
not my friend. Apologies for the typos.