Yeah, Bermuda grass is just one of those lousy things you need to pull. The best way I've found is NOT to pull it with your hand so you year a snap, because that means you left a big chunk of
root behind. Stick a shovel in and lift carefully, trying to pull up the
roots, then lift the grass and root off the shovel, drop it in a
bucket. I escort mine off the property! And while the grass is there it is using up all your great soil stuff, that's why it's so darn healthy.
One of the very best composts I ever had was mushroom
compost, so your Hugel bed is a really great one if you are getting mushrooms. Plants and seeds just love all those rotted/composted mushrooms/toadstools.
You might try really, really thick
mulch, like a minimum of 6" on the soil after you've gotten rid of the Bermuda. And keep it very thick, plant right in the mulch. But you'll have to go around the outer edge of the mulch with a shovel, like above, and remove any grass that starts to show up again. I read once where someone had some Bermuda grass come up at the edge of their garage slab, grow across the 20 foot slab. Never turn your back on it!!
An important distinction: Permaculture is not the same kind of gardening as organic gardening.
Mediterranean climate hugel trenches, fabuluous clay soil high in nutrients, self-watering containers with hugel layers, keyhole composting with low hugel raised beds, thick Back to Eden Wood chips mulch (distinguished from Bark chips), using as many native plants as possible....all drought tolerant.