Reduce Fizz from Palm Mutes

Why would palm mutes be any necessarily fizzier than the tone during any other playing? Doesn't make sense to me
 
It's mostly in your hands, experiment (like others said) with different picking/muting technique.

The guitar/bass/drums on that recording are not being played very tightly together - if you guys can tighten up your playing it'll immensely help the sound of the mutes (as you'll have the kick and bass reinforcing the guitar).
 
fizz in amps is the result from having their volume too low, you have to get the thing cranked high enough to get rid of the fizz for the amount of distortion that you want. Having the bias set correctly helps if the fizz is the result of crossover distortion. Only other advice I can give is that if you are recording, Low Pass filter to help if you need it, make sure you are not picking too hard/your action isn't too low (string-fret buzz), and make sure your gain isn't too high.
 
fizz in amps is the result from having their volume too low, you have to get the thing cranked high enough to get rid of the fizz for the amount of distortion that you want.

This.
To me this guitar sounds almost clean, you need to drive those amps, get a bit of beef and sustain into it!
 
do move the mic closer to the cab if it isn't set all the way to the cab, throw a high pass (60-80) low pass (12K), reduce some of the 200-300 area and maybe a slight hump at 10K. That should give it a more direct in your face.
 
I think there are a few things you could do to fix up your harmony guitars.

Put a tubescreamer type pedal infront of your amp. Dial in a "singing sustain". I prefer my old LP for harmonies but that's just me.
Experiment with the different guitars you own.
Boost around 10k.
Play like you mean it, as in good vibrato and a forceful pickhand.