I had worked for a weather radar company in China, and had designed X band LNA by myself.
Fr your questions, it seems not an issue. You know when the TX transmits, there is a protection before the LNA, and the radar would control the protection to work so that less power can be leak into the LNA.
Even your leakage is about 0dBm, it is pulse signal, not CW, you should make 0dBm divided by duty cycle. It is a coarse way.
Of course you can test your LNA with 0dBm pulse signal in various cinditions.
And you can also ask the LNA vendor to do so for you.
Another solution is designing the LNA by your self and you can select each stage amplifier so that you can control the IIP1. My LNA has three stages, so that I can design more freely.
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There is a protection diode (waveguide structure) before LNA.