The goats eat the poison ivy and then the area is treated. You could address your concerns with a place that rents goats and has expertise in this process, but you seem quite averse to the idea.The main issue is that renting goat will be a constant thing until the plant is so starved it does repopulate since the root is still there waiting to put out more leaves.
I’ve honestly never hated a plant or area so much before as I do living here and it’s realllllly hard to get over how amazing this place could be but the stupid ivy season ruins it
Anyway, good luck, I hope you find an acceptable solution.