The gardening service refuses to come out and take a look. What could the problem be and a potential fix?

Pictures as an example are below:

I think you could do some research on the different types and what damage you are likely to see as the heat of summer hits the grass. Bermuda is one especially that turns brown in the hot summer sun while St Augustine, given ample water, does especially well in the heat and stays green into fall.

I'm surprised the lawn company has failed you--be sure to let them know you are making a complaint to the Better Business Bureau of your county. The problem of the lawn persists though and if not consulting another expert, I suggest, at least, talking to your neighbors--find out if they are experiencing similar problems or if they know what you have. A talk perhaps with your county extension service might give you some expert advice on what the problem is and how to solve it. The extension service is found in your telephone directory under "county government"....or call such service at your "State" funded college. In this case maybe Florida State University Extension Service. Failing that, call your local 'full service' nursery and ask them about it.
This is the type of problem you shouldn't rely on well-meaning amateurs who 'guess' at what you should do.

From my view though, as you have stated the amounts of water that has been given the lawn, I'd start with that as a possible reason why your lawn has done what it has done.

Really? You think Bermuda turns brown in the hot summer sun? So all those southern golf courses with Bermuda greens and fairways turn brown in the summer? The opposite is true of Bermuda, it loves the heat and sun, cold weather turns it brown as it goes dormant.

What does that mean?? The grass is the standard grass everyone puts in Florida that wants the lowest maintenance and is the most resilient to the Florida sun, I forgot the name though. It's nothing special.

I honestly believe it rained so much at night that it overwatered my new sod and gave it a fungus problem. I've been treating it with corn meal for the past two weeks. No positive results yet, but I've heard it can take quite a while.

Your grass looks the same as mine, and Floratam is very popular in Florida too. The way you describe the heavy rains and sprinkler waterings you may have a fungus problem too.

Put some corn meal on it, it won't hurt it, and just may help.

Frequent watering is needed to help get roots established in sodded lawns. Your lawn was mature enough to back off on watering about 60 days ago. This time of year you should only be watering it 3-4 times per month. If you get rain, you might back off to twice or once a month.

I am the corn meal guy. I used to get that same fungus every spring. It will not go away by itself and can wipe out the lawn and all sod you bring in to replace the dead stuff. After I got the hang of corn meal I have not had a problem. If I get it, I apply corn meal and it goes away.

What kind of corn meal?
The same stuff you get at the grocery store. Most of us prefer to save money and buy it at a feed store. Cracked corn works, too.

How much?
Apply at a rate of 20 pounds per 1,000 square feet. Apply to the entire lawn because you don't know where it will pop up next.

When?
You can apply corn meal any time of day, any day of the year, rain or shine. Reapply at the same rate 3 weeks later.

Precautions?
Not really. Don't apply a chemical fungicide first and then expect the corn meal to work.

Why not chemicals?
Because you live in Florida. Check the label on the bottle and they want you to apply when the temp will not exceed 75 degrees over the next 48 hours. When is that? January.