Hello Kate, I had not heard of hoary cress and just looked it up: what a monster!
Its growth habit sounds similar to couch grass and bindweed combined with a high seeding ability, oh dear.
I am sure that mulching offers you more hope than cultivation but suspect it will be difficult to eradicate among the soft fruit, though easier to keep on top of.
Yes I would keep polythene/membrane on for a whole year.
And its not fair of the other allotmenteers to complain of your approach which sounds highly conscientious, diligent and will be effective.
Do keep us posted and best of luck with it.
