Best yard spray for bermuda grass to keep the mosquitos and flys out?

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You are going to find this suggestion strange and improbable. Some twenty years ago I found myself in a situation where there were pests about an enclosed courtyard of a college dorm. What to do to not expose the students to the risk? As Momtobe I leap to the conclusion you are in the same type of situation.
Regardless, fear not, for simple is both my name and solutions. Bermuda grass generally does not attract the bugs you describe. Standing water, however, does! That aside I want you to get a hose-end sprayer that is capable of being calibrated. You are also going to get a large container of Pure Ivory Liquid Dish Detergent.
Follow the directions from the sprayer, and set the sprayer at the equiv. of 1tblsp per gallon of spray. At Dusk spray the entire property, turf, flowers, shurbs, et al. with a spray for three straight nights. On the third night concentrate your spray efforts to any/all low laying areas or water accumulations.
Remove the accumulations of water through using sand or Gypsum to backfill. Do not use topsoil. Continue using the Ivory Soap solution for a month thereafter. If you are not seeing any results contact your local County Extension Agent for other non-toxic remedy. They will advise Safer or Bayer products.
I firmly believe you will get results by spraying at Dusk. This is the beginning of activity. It will take a few days to a week to effect the cure. If it does not work the worst you will have is a lawn that has been bathed and soaked in Ivory Soap. Could even get a commercial out of that. Good Luck