If you have a bermuda lawn, you'll always have bermuda in the bed.

What you might try next spring is setting out big planting pots on low pedestals of varying heights. Running a solid irrigation line with individual feeder lines (water)to each pot. Big pots might want two.
Then plant a ground cover underneath leaving spots for you to put walk stones.

This kind of elevated planting and direct watering to the pots will deprive the trees competing for water and once the ground cover (it can be as simple as decorative purple sweet potato plants mixed with maybe pachysandra or some other green that blocks out light from the wandering bermuda).
Mulch is good but not too deep and not pine straw. Ground bark is good but collar it at the bed edge to keep it from floating away. A thin collar of well packed pine straw works well and catches blown lawn clippings without looking messy.

It's easier to smother grass around (and add mulch around) taller plants with one main trunk/stem. A huge area like that can cost a lot to cover with new mulch often, so I would definitely supplement that with lawnmower-shredded leaves in the fall, and cut grass whenever possible.

As to root competition, that's a hard one. You could root prune behind the fence line. Do this when the soil is moist. And redo it at least a couple of times a year. Your bed has the potential of being fabulous.