It sounds like you are getting information from all over the place. The big hint was applying gypsum. Unless you know from a soil test that your soil is acidic, then you should not be thinking of adding gypsum. Generally Texas soils are already too alkaline.

Why do you want to aerate? Do you think the soil is compacted? Does it feel hard when you walk on it? Does it feel soft(er) when it is wet from rain or irrigation? If the soil feels hard when dry and softer when wet, that is perfect. If it feels hard when wet, then you could aerate it and that might help. If you do go ahead and aerate, then it sounds like you have done your homework. Aerate a lot more than you think you should. While you are at it, if you have high spots that you would like to lower and low spots you would like to raise, then hit the high spots harder and sweep the cores into the low spots to melt back into the surface. Generally you should keep the St Aug tall but if it is 4 inches high, that makes sweeping cores more difficult.

A really good alternative to core aerating is to spray liquid soap on the lawn. I hang out in other forums (which Gardenweb does not allow links to, for understandable reasons). In one of those forums they have developed a tonic (shades of Jerry Baker) that really does help soften your lawn. I tried it last year and it worked great for me. In fact it is still working for me. The idea is to break the surface tension in the irrigation water and allow it to soak down farther into the soil. Then that moisture down low holds the moisture content up higher for a longer time. The explanation drags on much longer but the bottom line is the soil is a lot softer. Okay, the mix is just soap. Their tonic is a mix of aloe vera juice and a special soap, but they make that stuff by the 5-gallon pot. I just sprayed generic baby shampoo through my hose end sprayer. Adjust the spray so the stuff coming out looks slightly foamy. Then spray. Following the spray you can irrigate. If you do this right before a rain, then you don't have to irrigate. Repeat in 2 weeks and again 2 weeks after that. That's all there is to it. I believe the soapy spray will save you the expense and hassle of the core aerator.

The key to choking out bermuda is to make the St Aug as dense as possible. For that to happen it needs water once a week in the heat of summer, tall mowing (highest setting), and fertilizer. I use organic fertilizer (corn meal specifically) so I don't have to worry about when I fertilize. With synthetic chemical ferts, you should not fertilize after Memorial Day. In Houston even Memorial Day is pushing it for the heat. If you decide to really go for it and want to make it as dense as possible, you can apply any organic fertilizer at a rate of 10 pounds per 1,000 square feet every month. 10 pounds is the lowest amount that makes any sense. If you really wanted to make an impression, go to 30 pounds per 1,000 every month, but that is almost asking for lawn mower problems. Pushing a lawn mower through very dense St Aug can be a chore.

If you want to apply compost, the application rate is 1 cubic yard per 1,000 square feet. NO MORE! Even that can smother your grass if you don't sweep it in with a push broom. Any grass that remains buried after you finish with the sweeping will not return, so get all those blades up and all the compost under the grass blades. Compost is the most expensive thing you can apply to a lawn. While it is good, I think it is over hyped. I would MUCH prefer to spend that money on organic fertilizer rather than compost. The bang for the buck is about 100x better with fertilizer.

You did very well in giving enough info. Thanks for doing your homework.

FWIW if you want the SA and kill the Bermuda off call a landscaper and have him spray the yard with Atrazine.

Atrazine is the most recognized herbicide to be used on SA. It is both a pre and post emergence herbicide. You can buy it in a granular form, but not liquid form as it takes a license to buy and use in a liquid form. In a liquid form sprayed on Bermuda will kill Bermuda. Wait until the Bermuda is actively growing and hit with Atrazine.

So can I buy Atrazine as a stand-alone granular herbicide from Lowes, etc. and apply myself? How do I know if I have Bermuda, Buffalo or some other kind of grass? Hard to identify going by web pictures, for some reason...

I am using gypsum since it was my understanding that it helped break up clay soils. My lawn is pretty hard in the summer, but does get a bit softer after rain.

Rather than buying I will recommend you rent an aerator, it will be much cheaper and you will not have to go into the headache of maintaing it. You can rent an aerator from some Construction equipment rental company.

Here is a link that might be useful: Construction equipment rental company

You can buy liquid Atrazine. Have used it several times. Here's a l;ink to buy Atrazine on Eb ay.

Here is a link that might be useful: Atrazine

Lowe's carries Scotts Bonus S that has atrazine in it. But i wouldn't apply it until Fall.