How do I determine the requirements of a switching FET?
The gate characteristics are important.

The total gate charge will dictate the size of the gate resistor you'll need, and subsequently the power dissipation due to drive. It will also affect the switching speed of the MOSFET, which means if you choose a replacement with different gate characteristics, you'll also likely have to play with the gate resistor values in order to achieve similar switching characteristics.

The gate threshold voltage will also play a role in the switching speed of the device. Make sure your replacement is in the same ballpark as the original.

Other parasitics (\$C_{iss}\$, \$C_{oss}\$) are important in soft-switched topologies, but for an ordinary half-bridge shouldn't be too crucial.

600V should be fine for a 460V half-bridge, since the FETs only see Vin worst-case.

Big-current MOSFETs tend to have low \$R_{DS(on)}\$ values, and are often chosen to minimize conduction losses. There may be high peak currents during abnormals (transformer short, etc.) which may make the part appear to be over-rated at first glance.

Calculating the peak current without knowing things like the transformer inductance, switching frequency, etc. can be hard - it might be easier to just stick a current probe in there and measure it (or measure across any current-sense resistor you may find).

Fast body diodes may improve robustness (assuming there aren't discrete diodes on the PCB in parallel with the body diodes) and wouldn't hurt in my estimation, so long as their current rating is sufficient. Again, this is more important for soft-switching topologies.