Is it possible to feed the HMC598 with only VGG and switch VDD using a P-Mosfet within 100nS?

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You should be able to set VGG1 and VGG2 to their recommended values and then target the Vddx to turn the part on and off. Just be careful how to handle the power up and power down, making sure that the gate voltages turn on first and turn off last. In order to achieve a 100ns turn on speed, the MOSFET will have to be placed between any power supply decoupling capacitors and the HMC598. You may find the PluserPlus reference design useful. It is a reference design for pulsing GaN PAs on the gate or on the drain( https://www.analog.com/en/resources/evaluation-hardware-and-software/evaluation-boards-kits/adpulserplusebz.html) . The current and voltage levels are much higher than what you need but the architecture will be similar to what you are doing.

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HMC598 : FAQ