I have more or less the same problem, but on my side is groundcover. My neighbors have St. Augustine. I've been keeping them apart for a little over 4 years with a regular spray of glyphosate (Round-up generic.) It actually works pretty well but results in an approximate 8" bald strip. One doesn't notice this from much of a distance, but it is not pretty (except to me in that it means I'm achieving my goal) close up. It is really not enough of an eye sore to distress anyone, including the neighbors or any neighbors. Down the center of the bald strip is the top of buried landscape plastic edging. I needed something to permanently mark the strip for tidy spraying but when I went to install, I discovered this edging there already, but buried. I just pulled it up to where the top could be seen so I have a guide. Each side gets about 4" width of spray. I am grateful that the edging penetrates about 6" into the ground and is a root barrier also as my neighbor's yard is full of weeds that spread by creeping roots. I really need a divider to extend vertically above ground, but this is not possible. It works pretty well so long as I don't miss spraying. Last year I was gone for several weeks, and busy also. By the time I got around to dealing with some rampant St. Augustine runners, some had gone as much as 7' into my yard! I vowed never to let that happen again!

At the other side of the yard, I have the same condition but different neighbor and a paver mowing strip (8" wide) instead of the top of 1" wide plastic edging. I spray (carefully with fan tip) the edges of my groundcover and neighbor's St. Aug. -- whatever portion hangs over the mowing strip -- and this looks much better with pavers than the other side of the yard with the bald strip. The pavers only penetrate into the ground 2 1/2" but this neighbor does not have the vile creeping root weeds.

At an earlier time in the back yard, I had to keep two groundcover beds separated (2 different types of plants) and created a brick mowing strip combined with aluminum flashing. The flashing, set on edge, penetrated 10" into the ground plus another 2 1/2" for the brick, for a total of 12 1/2" subsurface soil separation. It was a relatively low cost/low tech solution that worked well. I had to reconfigure this area and subsequently did away with it. The bricks were set on top of mortar as a way to hold them together and in line, and their joints were filled with sand at the surface. A taught string line guided their installation. Here's a rough schematic ...