I made a major mistake using Bermuda seed for erosion control in my back yard when I bought my house and there was no sod.
It is tough to eradicate. If you don't care to grow anything else any common Bermuda such as Sahara will do fine. If you like to garden, then reconsider.

Rye grass is most common for this purpose. Wait til temps are regularly in the high 70's generally Oct til mid-Nov is the best time for overseeding . Perennial rye is finer bladed and more expensive than annual rye. Both are commonly used. Tall fescue can also be used for overseeding bermuda.

For garden areas elbon rye is great for a green cover crop that is tilled in to the garden soil come early spring to add organic matter and nitrogen and keep weeds down over-winter.
Happy Growing David

Keep it in mind, bermuda will have terrible green up start when overseeded with rye. If it was me, I'd just not do it at all and live with brown grass for the winter. We're talking about maybe 1-2 months difference having completely green lawn between doing nothing and overseeding...

If it was me, I'd pull out all the bermuda and plant anything but, including weeds.