Bermuda rhizomes can be tough to kill off without chemicals. Took us well over a year to eliminate most of the Bermuda in our small back lawn organically, but still resorted to glyphosate spot treatments for patches that returned before we plugged with Buffalograss.

If you don't want to resort to herbicides, you could try hiring someone with (or renting) a sod-cutter to scalp it and spend the next season or three digging out the spots that will return from the deeper rhizomes. Whatever you use for mulch or groundcover in the area could also help suppress any return.

Also, some electric edgers have a "trencher" mode that might help you. We use a Black & Decker model to trench around beds in both St Augustine and Zoysia. They can occasionally be found used for less than $30 around us, so worth checking the local options to experiment before buying new.

Metal edging will NOT keep out Bermuda grass. It goes under, over, and thru the connections.

Sherry-

We were debating between having our masonry person install a low stone brick wall around the trees to separate it from the lawn. Our neighbors are having what appears to be success keeping Bermuda out of their planting beds, but they may have commercial grade heavy duty landscaping fabric in their planting beds that are edged with metal edging. What successfully keeps Bermuda grass out of areas where you do not want it?

Round UP will do it. We made the decision this year to reduce our grass area by 75%. Round Up worked great. Spray it once and watch it for several weeks. The stuff that dies will be obvious and so will the places you missed. Then spray the places you missed. I would spray back at least 10 feet from where you want to have zero bermuda. Then still keep an eye on the dead grass area, because we're seeing bermuda coming back 2 months later. Here's a pic.

At the very bottom (blurry part) of this picture you can see the bermuda sprouting.