#Regarding output capacitance and matching.
I would first breakdown the problem (so studying each (amplifier) stage separately).
Adding inductance to ground (as close as possible to the drain) will compensate this current, you may use L4 for that (make sure you use a good bypass capacitor(s) for C10 as it will carry reasonable RF current).
I would not recommend full conjugate match. Though it gives maximum gain, it may lead to slightly voltage-saturated operation.
#Increasing capacitance with increasing RF drain voltage.
You are correct that the output capacitance reduces with increasing DC drain voltage, but for the AC case it is different. The negative half of RF drain voltage experiences an increasing capacitance though the positive half experiences a reducing capacitance. The increase is larger then the decrease, so effectively, the drain capacitance increases somewhat.
In a parallel equivalent circuit, your 50 Ohms is shunted by a voltage dependent –j22 capacitance. Some change in this capacitance will introduce phase shift.
You may transform your 50 Ohms load to a lower value (as seen by the drain). This reduces the effect of the capacitances. This requires more RF current and it may become difficult to maintain class A operation. You may consider class AB operation and find a sweet spot (bias current issue). I know it reduces gain and it requires you to observe the impedance as seen by Gm for higher harmonics as now Id will contain lots of harmonics.
Do you have good familiarity with a simulation package (that allows non-linear simulation)? If so, you may change an existing transistor model (including the non-linear capacitances). It takes time, but in simulation you can turn on and off certain behavior to see the effect on phase versus output power. I learned a lot by doing this for a 100 MHz linear PA with a spice simulator.
To be honest, I don't believe that in your current circuit, the mosfet's voltage dependent output capacitance is the root cause for 30 degrees phase shift change versus power. It introduces a fixed phase shift, but as it is off-resonance, the effect on phase change will be small. It can be a secondary source (as you need to apply more gate drive).
Class D (or other switching scheme).
I reread you original posting again. You mentioned you only need about 6 dB change in power. In that case a switching class amplifier based on some medium power upper UHF ldmos can be an option. You only have to halve the supply voltage to get a 6 dB step. Note that such amplifiers are very load sensitive. Some inconvenient load and yo have to place a new mosfet.
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Unfortunately, all relevant components, except for the diodes are missing in your schematic. Seeing the paper, I get an idea, how the T/R circuit
could be connected. To understand possible phase shift issues, we should see the real circuit used in the test. Presently, it looks like a shorted filter output with no antenna feed at all.
I also have problems with this T/R switch, but when it contains quarter wave section not shown in this circuit, it may function.