Okay I got my shouting out of the way. Let the contractor grade the soil. He will leave you with the perfect surface for sod or seed. Tilling can give you a perfect surface with a lot of work but the problem is that the subsurface is uneven. It it virtually impossible to till the subsurface evenly. Then in 3 years when the fluffy tilled soil finally settles all the way down, it will settle to the contours of the uneven surface left by the tiller. In other words you will have a bumpy lawn. With bermuda a bumpy lawn is the worst thing you can have. The landscaper with a tractor, on the other hand, can use his ripper tines to dig the soil to one specific depth and do it evenly. Then he'll go over it with a box blade to level it. He might do both operations at once.
I can't think of anything I would do in your circumstance. Once the sod or seed is put down, the best thing to do immediately is roll it down with a water filled roller. This ensures you get a good contact with the soil. Roots cannot grow through the air so you need good contact. If they don't roll it and they WON'T do it, go rent a roller and do it yourself. It is important.
If you get sod, the first thing I would do (after rolling) is apply ordinary corn meal at 10-20 pounds per 1,000 square feet just in case the sod became infected with a disease either at the grower or on the pallet. It happens. The corn meal is a reasonably good fertilizer, too, so you can't lose anything by using it.
New sod needs water just like new seed. You need to water a couple times per day, every day, for 2 weeks. Then start to back off on watering until you are watering no more than once per week. You'll have to water a lot longer (deeper), and that is the idea. Deep infrequent watering is the key to good turf grass.
Mulch mow at your rotary mower's lowest setting for bermuda. If you have a reel type mower (preferred for bermuda), start at 3/4 inch high and see if you want to go lower. Bermuda takes on a horizontal growth habit when mowed below 1/2 inch. Golf greens are mowed at about 3/32 inch and you know how nice those are. But those are high maintenance requiring mowing every day to be that nice.
If you want to apply compost, do that after you are sure the roots are knitted into the underlying soil. The application rate for compost is no more than 1 cubic yard per 1,000 square feet. You have 2,750 minus the driveway so keep that in mind. 2 cubic yards should do you. You apply that by piling it up into mounds across the lawn. Then use a shovel to fling it out and about. Then use either a leaf blower, push broom, or the back of a hard rake to get ALL the compost below the level of the grass. I've seen beautiful bermuda lawns smothered for years by one goofed up application of compost. When you are finished you should be able to see all the grass and no piles of brown.
With bermuda you can fertilize every month for a dark green lawn or you can let it go unfertilized. Unfertilized bermuda is yellower and less dense. As a minimum I would fertilize with the best organic fertilizer (in my mind that is pure soybean meal or corn GLUTEN meal) on the federal holidays. Memorial Day is next followed by 4th of July, Labor Day, and Thanksgiving. I would not skip Thanksgiving even if the grass is already dormant. You're not feeding the grass with organic fertilizer, you are feeding the soil microbes. They are still there when the grass is dormant. If you want to do the compost and fertilize on the same day in May, that should be fine.
Where do you live?
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