The newspaper blocks the light and that kills the grass. Here’s a How To article: https://dengarden.com/gardening/How-to-Kill-Grass-Without-Using-Chemicals

The active ingredient in Round-Up, Glyphosate, is not harmful to nearby trees. It is taken in by FOLIAGE and translocated to the other plant parts, killing the plant ... but leaving the tree alone since you didn't spray it on tree leaves. Also, it has a relatively short shelf-life in and on soil, breaking down into harmless substances. It doesn't go on killing forever. But the light-smothering technique will work fine. It will just take longer.

What you're doing by chopping up the Bermuda with a hoe is ... planting more grass. Each piece remaining in the soil will develop new roots and start growing again.

That is, frankly, a bizarre bed shape, looking like it belongs in a mini-golf course. You might consider redesigning the bed shape while you're working on killing the grass. A circle or other simple shape would work much better.

Considering the size of the tree trunk I think that bed is too large. Is having a bed here something you really want? The shape is bad too.

Spray with Roundup and reshape the bed.

BTW, bermuda does not need to be killed just once, but it usually takes two or three times as it recurs, as there is a lot of energy stored in its roots. I would not rush to replant the bed, but observe it for a time and make sure the roots are killed. With the cardboard method, this takes time anyway.

Don't use glyphosate. Douse the green with strong vinegar, cover with newspaper and mulch. 6 weeks you'll be all set

Round Up isn’t harmless. Lots of studies and research on this. Especially harmful to the bee population https://www.weedkillercrisis.com/topics/roundup-weedkiller-vs-bee-populations/

No pesticide is harmless....that's why they are called 'cides' (Latin, meaning killer). However, they do have value in today's world, when used in moderation and according to label directions. And glyphosate/RoundUp is among the safest, despite all the hype one can find online about it being so toxic (none of these reports are fully conclusive). One should never use any pesticides when pollinators or beneficials are present. But since pollinators have no interest in grasses, using it for this purpose is especially well-called for and not likely to be of any harm.

You have a longer term problem. Whatever method you choose to kill the bermuda in the bed now - what will you do in the long term? Once you replant the area, how will you keep the bermuda out? I don't know if a pre emergent will deter the bermuda roots from infiltrating.

Well, it's not as entirely hopeless as all this sounds! If you want a planting around the tree I'd make the area larger, dig out all the grass and roots as you can, dig in some compost to improve the soil. plant it, give it a thick cover of mulch, and when Bermuda grass pops up give it a small spray of Ortho Grass b Gone.

I don't like using "cides" of any kind, but since I have a mixed lawn of St. Augustine and Bermuda grass, which I didn't plant that way, but the Bermuda blows in, I have no choice.

Grass b Gone doesn't harm broad leaved plants and kills Bermuda. It takes about a week for it to die, but die it does. I use G-b-G very sparingly, getting the nozzle close to the grass before spraying gently while holding my breath.

After the bed is established I have very little problem with Bermuda infiltrating it. A brick, stone, or metal border and thick mulch helps keep it out.

The problem is the bed is surrounded by Bermuda. I have this problem as well. My best suggestion is to postpone the flower bed, but since you have started plant St Augustine grass there instead. really indulge it with plenty water and fertilizer. After it grows and spreads it will completely choke out Bermuda grass. the come back and cut out the St. Augustine in the bed size you want taking care to leave the St Aug. a wide margin to keep Bermuda grass out. st. Augustine can be smothered with cardboard and compost mulch way easier than Bermuda grass.

I don't use any poisons, and to combat my couch grass problem (Bermuda Grass is the same thing I think) I used layers of newspaper. I found the method fabulous.

The newspaper layer physically prevents the beast from growing through. Without light, it cannot survive.

You need a little patience, but it's nice to feel as though we are not contributing to all the poison that gets sprayed and splattered about.

Sometimes a small piece of couch/Bermuda grass pokes it way up through the layered layers of newspaper. Fear not. When this happens, resist the urge to pull it out as that will tear the newspaper layers and allow more of the c/B grass to show their heads. You can simply cut off the offending pieces.

Good luck.

While you are at it, consider increasing the size of the bed. You can't get much in there right now. Definitely cardboard. I've been doing it and mulching over the top with a lot of success.

Even if you are successful in removing the Bermuda grass from the planting bed, you will be forever pulling it out of the bed as it surrounds the bed and WILL creep back in.