Your best idea is the last half..baby shampoo and alfalfa pellets for soil compaction and water retention and just letting the Tiff/Bermuda naturally overtake the annual rye grass.

You mow as soon as it need mowed. The taller the Rye gets, the more nutrients, water and sunlight it robs from the Fescue seedlings which have not germinated yet.

Oh man. I thought I had the answer until you threw in the dogs. Generally, but especially in your climate, you should NEVER overseed bermuda with anything. You can't kill the rye in the spring. All you can do is hope the bermuda will get enough sunlight when you mow to rekindle for the new season. My next door neighbor seeded rye into his St Aug last year and still has bare spots where the rye choked out the St Aug in the spring. Bermuda needs all the sunlight it can get to restart. Even the thin little rye blades will delay coming out of dormancy. But you have dogs. You are danged if you do and danged if you don't. In your case I'm not sure you won't still have a mud hole all winter but I'd try it anyway. It can't be a worse mud pit. Would you consider replace the back yard with a nice, deep mulch? Dogs love it, it stays pretty clean, weeds can be pulled out easily, no mowing, it builds the underlying soil, no watering or fertilizing, chipped tree limbs are free, and no mud.

the lawn at my office building just rocks all winter long and just wonder how they do it. We I am a sod farmer just north of DFW area and also own a landscaping biz that does a lot of commercial properties in DFW. We have a secret to over seeding Bermuda, actually two secrets. 1. We use a blend of Rye and Fescue. 2. We use a herbicide early in the Spring to kill the Rye and Fescue off so the Bermuda will not be deprived of water and nutrients thus delaying green up and speedy recovery from its winter slumber. Otherwise it would be July before the Bermuda recovers, and two months later you would have to over seed again futher weakening the Bermuda. It is a viscious cycle you really do not want to start.