Input cap of power amp affecting power supply loop compensation?
You never have to just guess and check.  Or at least, you can make educated guesses, and then carefully simulate.

You can estimate the input impedance of the TPA3221 by observing two things: first, it's a switching amplifier, so power in = (power out)/efficiency; and second, it's got those great big input caps.

So, let \$\eta = 0.9\$ be the efficiency.  Then \$P_{in} = \frac{P_{out}}{\eta}\$.  For any given power level, if you ignore the capacitor, \$V_{in} i_{in} = P_{in}\$.  Do a bit of math, and you get \$i_{in} = \frac{P_in}{V_{in}}\$.

This is a nonlinear equation, so linearize it around some operating point.  Call it \$V_0 = 30\mathrm{V}\$: $$i_{in}\left(V_{in}\right) \simeq \frac{P_{in}}{V_0} + \left(V_{in}-V_0\right) \left . \frac{d}{dt} \frac{P_in}{V_{in}} \right |_{V_{in} = V_0}$$
This gives you an effective input impedance of \$R_{in} = \left . \frac{d}{dt} \frac{P_{in}}{V_{in}} \right |_{V_{in} = V_0} = -\frac{P_{in}}{V_0^2}\$.  Note the negative resistance -- this will not help stability.

With the cap in parallel smoothing things out, you can use the average output power to calculate \$P_{in}\$.  The worst-case negative resistance will be at the highest power output.

On the supply side, if I didn't want to actually do the math to calculate the output impedance of the switching stage, I'd simulate it with a current source as a load.  I'd use a source that's drawing the correct DC current to make \$P_{in}\$ to the following stage correct, and I'd add a sinusoidal current (less than 10% of the standing DC current) to it.  Then I'd look at the sinusoidal component of the resulting output voltage.  Do this for a few frequency points, and you should be able to get a model of the output impedance of the power supply stage.

As long as the sum of the parallel equivalent conductances of the power supply and the load is positive at all frequencies, then the result should be stable.

As an alternative to calculating the output impedance of the power supply, you could make a constant-power load in SPICE using a dependent source.  Put it in parallel with those \$470\mu\mathrm{F}\$ caps, then just simulate your system and look for signs of instability.  You'll need to make your load in a way that it'll allow the supply to start -- you probably want to have it be constant-power up to some maximum current, so it's not trying to draw infinite current at zero volts.