Your thoughts about the sound I will get from this make me think I'm not on the right track. Most likely I don't know enough about circuitry and electronics at this point to know what I'm trying to ask.
What you've asked for help implementing is perfectly clear. I did preface the explanation with it being a bad idea, which is why you aren't seeing posts about this.
You're going to need a test rig, preferably a breadboard. Once you start experimenting with more than changing a couple components on the board, it's a good idea to have a method of altering things that doesn't require desoldering parts. In this case, though, you can do what you want without a breadboard. You just need some alligator clips.
First, forget the pot for the moment. It's just making things more complicated. (In the meantime, you can read RG Keen's "The Secret Life of Pots" to get a better idea how they work.) Grab a resistor -- let's say 10K (if this doesn't work, use a smaller resistor, since I don't feel like figuring out the LT1054's effect) -- and your multimeter. (You do have a digital multimeter, right?) Take two alligator clips and clip the resistor between them. Like this>
dangling lead that'll go to your power supply pin ... clip > resistor > clip ... dangling clip that'll go to your circuit.
Put the 10K resistor between your circuit's voltage input and the appropriate pin of your power supply or battery. Take another alligator clip. Its got two clips on it.
Grab your multimeter, put it on continuity. Touch one lead of your multimeter to one of the clips, and the other lead to the other clip. It should beep. You're probably saying, "Duh, Jon, of course it beeped, it's a wire with clips on it, we just used a couple to hook up the resistor." Well, a switch is just a couple wires with an actuator to change which wire you're using. But we don't need the switch yet.
Your mission, should you choose to accept it, is to hook up your alligator clip so that the 10K is no longer in the circuit, and then LISTEN to what happens when you put it back in the circuit. And back out of the circuit. And back in. Do it with some other resistors, too. Try 1K, 4.7K, 22K, 33K.
Maybe there should be a capacitor on the potentiometer or something. The sound I'm looking for is pretty raunchy, but wouldn't pop or contain any popping artifacts.
Leave the resistor in the circuit. The 10K might work, if it doesn't, use the 4.7K.
Alligator clips or a breadboard make it really easy to see what happens here. You can put capacitors in series or in parallel with the resistor. So grab your capacitor and multimeter. Clip the capacitor in series with the resistor. For this experiment, it doesn't matter what the value of the capacitor is, which might be a clue to some of the questions I'll pose.
How does the sound change? Measure your voltages and compare them to the other voltages you wrote down. How have the voltages changed? Can you figure out why? (The Wikipedia article on capacitors might help.)
Now let's try in parallel. Clip the capacitor so that its two leads are connected to the two leads of the resistor.
How does the sound change? Measure your voltages and compare them to the other voltages you wrote down. How have the voltages changed? Can you figure out why?
I haven't discussed the switch. The purpose of these experiments is to show you that while your idea has merit for altering the sound, you will probably discover that a momentary switch is not going to be a practical solution. I've been focusing purely on the pop that occurs when you suddenly and drastically change the bias of a fuzz, but a lower supply voltage also produces a lower output, so you have volume differences to contend with as well (and there is no solution whatsoever that I'm aware of in any fuzz where you could avoid a similar problem). My advice, though you probably aren't going to like it, if you want to momentarily is to build a complete second fuzz, and then use a momentary switch to flip flop between them. A switching scheme should be around somewhere for that, if not it would be a really good exercise to work it out on your own.
I'm going to append an explanation for the popping or scratching sound:
This is actually related to how a transistor works in the first place. A transistor is a transfer resistor
(this is a portmanteau, a word that combines the sounds of two other words, and the usage of that
word comes from Charles Dodgson, believe it or not). When you put voltage or current (depending on the type of transistor) on one of the pins, the resistance between the other two pins changes. Do that gradually at a nice musical frequency and you get a nice musical frequency out the other end. Do it really slowly and gradually, and you get a low frequency noise out the other end ... sounds like scratching. Do it a whole bunch at once and you get a pop. It doesn't matter where this happens, audio circuits don't particularly "like" it. It's commonly a problem for guitar pedals for the input or output to cause a pop when switched -- the pop ultimately happens for the same reason