After reading up on your link I’m more worried about that product damaging our trees I want to keep the forest but eliminate the poison ivy
Maybe hire goats to eat all the leaves, then put down mountains of thick paper or cardboard mulch, covered by mountains of wood chips? That might be enough to keep it buried so the poison ivy dies instead of resprouting. And all of that will eventually rot and fertilize the trees. The trick is to get it thick enough that the unwanted plants die before the mulch rots.
I mostly worry that most goats come with dogs and I have a pack that is not friendly to other dogs and I know dogs that protect goats would easily take on my pack
Depending on how many days it takes, you might be able to just keep your dogs in the house, and take them out for bathroom breaks one at a time on leash. Or board your dogs somewhere for a few days?