If it's the native species, it isn't invasive. (natives belong, so they can't invade).
Mowing isn't always effective for Opuntia, since, depending on conditions, the cut up pieces are quite capable of rooting and growing more plants. Also anything that doesn't actually remove the plants physically runs the risk of leaving spines in place to get you down the road sometime..
Hugelkultur would work- they will not grow up through a thick layer of wood and soil- as long as you will never be sticking your hands into the layers that have the cacti (see previous note!)
The plants grow from seed as well as rooted pieces, but neither way is fast to establish- if you dig up all the pieces they can't grow back any time soon, it will not come back from roots, and seeds take years to reach any size.
Definitely if you could waterlog the area, cacti do not like prolonged wet, but you'd still need to get rid of the spiney corpses.. The only thing I can think of getting rid of the plant entirely without digging might be a long hot fire..
Why not advertise for local cactus lovers to come rescue them? Many people treasure their native cacti and have beautiful gardens built around the locals and many other species.
