Took me a while but I figured out the fast strategy. It's simple once you know it...
A series of hints:
First, the really big ship.
It has a terrible weakness, and once you kill it, it won't come back for seven minutes or so. Yes, you can kill it.
You need to do what you do *twice*. So if you had any clever ideas but they didn't work, make sure to try them two times. (There is a good reason for this.)
Ships die if you hit their bridge. Of course, the bridge of the big ship is rather well protected...
So you have to ram it. On the *inside*.
Spawn your HQ right inside the big ship. The first time this will make a hole in the struc. The second time the big ship blows up. Your own HQ blows up too of course, both times, so you have to wait out the timer.
Of course, that's not enough.
You'll find that if you start lathing the things around the progenitor, you won't make it in time. There are several ways to deal with this, but only one that gives you the best time.
You *have* two more lathes, beyond the one on your HQ Already built and ready for action.
The combat lathe ship, of course. It's at the left of the map, with a full tank. It's strong enough to make it to the progenitor if you guide it well. (Yes, you have to actually fly it there, don't destroy and rebuild.)
Go via the north half of the map, keeping it over islands as much as possible. Particles won't hit it there as they're still miring the land. Fly in between the flip emitters, but avoid the large chunks of mire at the cannons. A good route will make or break things here.
Realize this cost the ship all its energy, at least part of it spent on lathing bits of random things on the way. Try again but turn lathe and weapons off before you start flying

Now, take the energy mine nearby, and then have your HQ and the combat lathe ship lathe the progenitor. They'll have to lathe two of the emitters there first before they will touch it. I expect you can beat my time with better management of the energy in the lathe ship.
I love the strategy for the big ship, first time I've seen it. 10/10 would learn a new thing again.
Now, to do it the "intended" way...