It sounds like you might have to uproot and re sod. Is it possible to set up a boundary of gravel or attractive rock formations between you and the neighbor's grass?

For St Augustine, which only spreads across the top of the soil, we were successful with a metal landscaping barrier. Of course I had to edge up against the metal barrier every other week.

Crabgrass killer controls centipede as well.......and Its safe to use on Blue/ fescue lawns. The usual crabgrass killers you see on the shelf (Ortho, Bayer advanced etc. ) are toxic to centipede, St. Augustine, paspalum , and other grasses that have that type molecular structure in the leaf.