Living in Tennessee and being in the transition zone is confusing.
You get alot of advice from neighbors and even so called proffesionals that are just wrong or more related to a cool season market.
Nashville and Knoxville are predominantly cool season markets, Memphis is a mix. Most Lawn and Garden companies dont even know how to deal with my lawn. I priced aeration of my lawn last week and the guy looked at me, in my face, so seriously, and told me that i was making a mistake aerating my lawn and not overseeding. Needles to say I will be doing my own core aeration this year.
Hey wait, did you talk to the same guy. lol lol lol
Bermuda lawns traditionally do not need overseeding. If you have very large patches of bare spots I would honestly get a pro plugger and transfer plugs. That way you are sure to get the right grass growing.
Also if your not sure what kind of bermuda you have right now you will have a patchwork of different colors and texture (leaf blade sizes)if you choose the wrong one. You may not think you care but you will. (or your wife will make smart remarks like "why is that grass a different color?")
If your lawn is thin and week then you need to take pictures and it can be fixed.
Overseeding bermuda will in my opinion be a big waste of money. You will get poor germination. And the cold weather varieties available by seed are not cheap. ( look at the price of Yukon Bermuda Seed) Why would you pay that much for a bag of seed thats almost guaranteed not to grow?
Now if your trying to do a complete renovation it is hard but in my opinion worth it. I am in nashville and my yard renovation was an ordeal. (understatement) But right now I also have about 75% green up. and its Mach 2nd. But that is a trait of my seed selection.