The Bermuda should not suffer from a January application of Roundup. Maintaining healthy turf is the best weed defense.

Many say if you're gonna use round up, to only spot spray and keep it to a minimum. Problem is that you are risking killing any bermuda that isn't completely dormant. Winter weeds will die off once the heat comes in. I would spot treat and pull as much as you can. Come spring, conduct your spring scalping, and then maintain your bermuda as low as you can, which will encourage it to become more dense and will fill in any dead spots.
Hope this helps.

Thanks, iriajs2009. In this case I have a pretty persistent, and extremely widespread, infestation of perennial weeds which spread by deep rhizomes (they would be as hard to pull by hand as Bermuda itself). Weed-B-Gon didn't control them, so I'm really hoping to be able to nuke them with Roundup. Spot spraying isn't an option, they are like literally half the lawn.

This spring is the first time I started maintaining the lawn seriously as well. I aerated, limed, fertilized and put down pre-emergent. This Bermuda has been living in pretty suboptimal conditions for a while, hoping the improving conditions will also help it fight out the weeds.

First I wanna start off by saying, have a lot of PATIENCE. Patience is key unless you start all over.
Answer the following questions and I will help you out where I can.
How low are you mowing?
How often?
Have you had a soil test done?
adding lime where it is not needed will do more harm than good.
Preem is only good for PREVENTING new weeds.
Where are you located?
I'm in Houston tx and my bermuda is still growing.
Can you name the weeds that you currently have? You mentioned perennial weed, do you know which ones?
Have you heard of celcius herbicide? I just bought some but it has good reviews on controlling weeds on bermuda during the summer heat.

I just completely read through your first post. Not sure how the weather is in your area right now, but once you get some pretty good freezing temps, then your bermuda should be mostly dormant if not completely. If I were in your position, I would pull the trigger and use round up in january. It will take care of cool season grasses and weeds. Once it warms up, we'll work on getting your bermuda to spread and fill in. I strongly recommend getting on a preem program and sticking to it.

In addition to iriasj's questions, I would like to know your watering regimen - how often do you water and for how long?

Also in addition to the one fertilizer app last spring, how many other apps did you do?