I see no reason why you can't use round-up on the suckers that pop up on YOUR side! We removed a Liquid Amber tree that suckered like crazy even after it was removed. Took one year fighting those suckers, but the big root is finally dead. By rounding up the suckers on your side, a little round up will travel to the main root. Could be a slow death but those suckers belong to you.
We had a neighbor with a beautiful shade tree on his side of the fence, long before my fig days, and we designed a gorgeous shade garden for our side of the fence. The neighbor moved and the new owners cut down the tree and ruined my micro-climate! You just can't control what a neighbor is going to do.
Suzi