About the heat of the matching circuit of MRF151A
Ok, so firstly a ferrite core inductor in the drain supply is almost certainly NOT what you want, go with an air core coil instead the losses will be much more reasonable. Usually I do an air core here, with a deliberately lossy ferrite the other side of the bypass caps (That will carry significant current and can usefully be ATC or similar), sometimes I kill the Q of the ferrite inductor with a few ohms of parallel resistor to reduce the tendency to low frequency oscillation.
You probably want to shunt C302 with a few uF in series with 10R or so again to damp the low frequency tendency to honk.
I am not sure how you are getting that much bias, given that R313 is making a voltage divider with R307, might want to look at that. Might also want to re do the bias generation such that an open circuit wiper contact on the trimmer will not be an expensive disaster. On the subject of bias, where is the thermal compensation?
Why do you have a directional coupler there?
The drain impedance will be both nothing like 50R and the signal will have a mess of harmonic content, I would not expect to get anything meaningful out of a reflectometer bridge in that position, it should go after the output matching and LPF networks. The heating is probably dielectric losses in the main, the harmonics are more then high enough in frequency to heat the dielectric on cheap FR4 significantly.
Incidentally, why is your output a tuned network? The usual approach in a band II amplifier is to go with a transmission line transformer (And usually a push/pull stage to get the useful reduction in second harmonic, makes the filter MUCH nicer to design).
Pro tip for RF devices using that package, bend the outside few mm of the leads upwards before soldering the device down, makes removing it much less sweary.