Dealing with unwanted grass by Blackberries & Rasberries

Any and all advice welcome.
Three years ago I planted one grape, one rasberry, and one blackberry (small bushes/vines) along 50 feet of fence at the rear of my yard. The grape and rasberry have grown little / only slightly. The Blackberry has propogated into some 30 plants via underground runners (so much so that I have moved 10 to my side fence).
The grass I once wanted along the back fence is now a nuisance. I have to cut it by hand with scissors or small sheers because an electric weedeater would also cut down the blackberry vines that produce delicous berries.
So what is best to give me a well groomed look but be low or no maintenance. If I have to choose between the two on this particular subject, I prefer low/no maintenance over well groomed. The area I am talking about is a 2 foot strip approximately 50 feet in length. Here are my thoughts so far. 1) I am willing to hand pull or dig the sod once (probably take me a while). 2) If I were to put down felt to prevent grass re-appearing I perceive I will stop the blackberries which I love from further propogating. This seems bad as the blackberry canes that produce must be cut to the ground before winter/in fall, and new canes replace them via underground runners. 3) I could put down real or rubberized mulch but perceive the grass would just spread/grow through in about a year (creating more work to clean up the area if I used rubberized mulch). Regular mulch I could leave and add to next year, but perceive the grass woud just start spreading and growing into it in a year. 4) Round-up on the grass would likely kill or affect the blackberries.
Any ideas welcome. Thanks in advance.

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