Can only answer as to how I would proceed.
Yes, remove as much as possible and cover. No reason why not to apply your compost under the cover. It will warm the surface and get the worms busy!.Its quite likely that the marestail will reappear. Its one of a few plants that predate the dinosaurs!!, it appears in coal measure profiles!, its a survivor. The covering should exclude light; therefore reducing its ability to photosynthesise.
Remove cover, pull out all evident shoots, thereby weakening it.
Recover.
Repeat.
Persistence will let you win-eventually!
I fear to comment regarding your expressed attitude to animal manures, only to say that you effectively are working with one resource short of a set.
Good luck
Cleansweep
