Plowing under will not work the pieces regenerate plus all the debris will spread into the air. From 1styou hand experiance I have can tell you plowing
under or tilling is the las thing you want to do. Goats would be a good idea. If your worried about long tern destructiveness just sell them or slaughter them once thier job is done. Other than that hand removal and turning the poison ivy into biochar would be what I would do.