No, do NOT ship birds in this heat! It may be ok to one post office to another but sometimes there is a delay during the terminal and who knows how long the birds will be sitting there.
If I were you, to get the birds in good shape, I would advise you to wait until cooler weathers. Some airlines have rules that no animals will be shipped out. Unless you want to go strictly on Pet Airways which they specialize any pets including birds to ship.