No it won't work as its a LINEAR AMP.
As soon as you reduce the voltage it will cease to function!
You need Class E which is way more efficient, simpler, cheaper etc etc.
They use those devices you quoted as they are modern, i.e. available, are cheap and are very easy to drive and have a low on resistance. There's no need to use GaN as they're for much higher (RF) frequencies. No real need to use SiC either unless the voltages are very high. Any crappy ol' FET will work as a PWM output as the frequency is only 200KHz or so....you just DONT want Ohms of on resistance as with highish current flow you'll drop volts and generate a lot of heat. SiC FETs can be use with high voltages keeping the current low, I2R is what generates loss.
With a linear amp you're best directly modulating the DDS and amplifying that to 3W to drive the linear. If you check my old posts I mention directly modding a AD9850 and there's a good article on the web about such..
Bought the same amp as you to do said build! You may beat me to it! Was also going to make a 3W SSB generator to drive it too!
Keep us posted!
Str.