Push-pull class E means to operate two single ended stages with a 180° combiner, transmission line transformer or similar. Advantage is cancellation of second order harmonics.
Sure it's possible to make a push pull class E. If you truly want the defining characteristics of class E (switching when V=0 and dV/dt=0), then the design equations will be completely different from the standard single ended class E.
If you don't need a "true" class E, then you can consider the current mode class D, which is a push pull switchmode topology, but doesn't satisfy the dV/dt=0 switching criteria. 400MHz at 100W shouldn't be too hard using GaN FETs.
I don't say they don't exist but I never seen, not even a simple schematic, of a high power, high frequency (>100W, >400MHz) Class-E, Push-Pull PA.
I've seen Class-E Push-Pull PAs working at 13.56MHz and 27MHz, but not at 400MHz.
Yes, the max power you can get out of a single device is highly dependent on frequency, mainly due to parasitic inductances (especially the source/emitter). To get CW 100W at 400MHz, I'd guestimate you'd need 3-5 amplifier cells, depending on what means you use to cool the devices.