This is a good time of year to grow grass, but Bermuda goes dormant with the cold, so the survival of seedlings might be in question. Why don't you instead focus on removing the weeds during the grass' dormant season?
There are a couple of products to try (in addition to the old-fashioned mechanical method of pulling the weeds): Agralawn
Crabgrass killer seems to take out more than just crabgrass. The featured herbicide in this is cinnamon.
Vinegar products have gained a lot of traction in the weed-killer marketplace and are combined with other things (like d-limonene or orange oil) to pack a punch. Here is a
more comprehensive newsletter with information about several of these.
Weeds typically grow during the cool season also, so if you can 1) knock out the existing plants and the 2) prevent the seeds in the soil from sprouting using something like corn gluten meal (after the existing weeds or gone or it will simply act to fertilize them - timing is everything with CGM. Think right around New Years, before you see new weeds sprouting in the lawn).
I'm sure the regular moderator of this thread will drop by at some point and offer targeted information about how to green up your Bermuda, soil and fertilizer treatments, etc. For now, though, you can plan your attack on the weeds as soon as the Bermuda turns brown for the winter.
Good luck!