Should I try an even larger cap value?

You could, but that doesn't sound like the problem.

How are you observing 0v on the drain?
Can you determine time duration of the 0V?

 

I picked up some 2N7000's and bread-boadred it, and the circuit you designed works perfectly for my needs! It's a hack, but forcing this display to match the analog switch will be incredibly helpful to my users. Thanks!! I really appreciate your time and effort! Cheers.

 

Actually, I do have one more question/favor: will the circuit work if VCC is dropped to 3.3V? Jow would that affect the values of the components in the circuit? Thanks again for your help!

 

will the circuit work if VCC is dropped to 3.3V

It may not work, since the Vgs threshold voltage of the 2N7000 is near that voltage, but you can give it a try.
If it does, you will likely need to increase the value of the capacitors to maintain the output pulse time.

Why do you want to use 3.3V?

 

It may not work, since the Vgs threshold voltage of the 2N7000 is near that voltage, but you can give it a try.
If it does, you will likely need to increase the value of the capacitors to maintain the output pulse time.

Thanks - that was pretty much what I was curious about, how great a voltage differential was necessary for this to work.

Why do you want to use 3.3V?

I was looking at the physical format of the PWM meter and noticed that a daughterboard directly below the meter would make it easier to position my leads for the "-" legs of the three tactile switches. Such a board would also have very easy, direct access to 3.3V, and I thought this might enable a more efficient circuit layout, but I don't want to get greedy. I can definitely make it work on 12VDC.

Thanks again for your help - this is a real problem solver for me!