Bruce, first question ... how long have you lived in Florida/had a warm season grass?

You *ARE* aware that warm season grasses go dormant/brown during their winter period, right?

Pets urinating on grasses can sometimes cause those kind of brown circles. Possible?

If that could be it, do a search on this forum for solutions to that (some add sugar and water it in, try to keep pets better hydrated to lessen strength of urine, etc.)

I know from nothing about southern warm-season grass types but from the picture, that does not appear to be damage from pet urine. That damage is more pronounced in a small area--it kills the grass roots. The picture shows more of a widespread area.
Question: Do you have a dog and does he pee in that area...if no pet, then you can discount that as the cause.

Dormancy...does warm grass grass types go dormant in small areas or is that a more widespread action...browning all over.
It started and you thought maybe a bug was doing it. You should never apply an insecticide on something you don't know what it is. If you think you should, then use soap and water...that will kill most bugs you can see. Then rinse the area well. If something else, try a insecticidal soap.
Fungicides....before applying, take a piece of the sod to your local full-service garden nursery and ask advice.
Is there a fungus that causes such damage.

Have you tried digging up a small section, see if there is any hiding bug that is devouring your grass roots.
Speak to your neighbors, have they had similar attacks...now or in the past. What did they do.
Have you tried giving your lawn a good soak.

As for giving grass a herbicide such as a glyphosate -- you should be sure the grass IS DORMANT before applying what can kill the grass. Evidently you can aplly "RoundUp" to dormant Bermuda grass with no harm done....but read the directions on the package/bottle and follow it to the T.

Just a shot in the dark but is it possible the grass was burned by muffler exhaust? Has there been a car or piece of gas powered equipment operated right there? We had a guy burn our grass with his pressure washer. It looked very much like that. It seemed to grow, too. I guess the grass was affected differently based on how far away from the heat it was. But it was all affected and eventually browned.

Since you have applied both fungicides and insecticides, I would go one more step and suggest you add a very light dusting of compost to the area to revitalize the soil microbes. For that area you would need at most a gallon of good compost.