OK, do not put a full bridge to replace D6,7. You do need a snubber across the mosfet say 150pF 1kV, and 47 ohm for starters (2W), also a string of zeners across the mosfet, 750 volts worth, say 5 x 150V 1.3W in series, and snubbers across D6,7 say, 1nF 500V, and 10 ohm for starters (2W).
The 1N5399 is too slow, you need 2 x 600V U-fast in series (<35nS Trr).
This should get you going for starters, the two pri side windings on the Tx should be bi-filar for good coupling. You should repeat them on each side of the sec wdgs to get the leakage down to reduce the snubber requirements.