Follow up on my repair:
1) Turns out I didn't need to replace the TLC2262 at all. The only problem I have is the broken Ratio rotary switch. By virtue of its being broken, there is no output when the footswitch is engaged. (The other telltale symptom is that the LED graph at the top lights up completely right at power-on and stays lit up the whole time, never dropping down.)
2) I desoldered the Ratio rotary footswitch with only moderate effort using this trick. Using a Hakko 808 power solder sucker, I completely removed the solder from the ten small pins so that they were standing naked in their holes. This means only the two larger housing mounting clips needed to be desoldered to remove the entire switch. I couldn't suck all the solder out of the mounting clip holes, so what I did was fill them in with leaded solder and heated the solder up by holding the iron there for a while, then I tilted the switch housing to pull the mounting clip out of the heated solder. Repeat for the other mounting clip on the other side, and the whole switch lifts out pretty easily.
I didn't bother with the Chip Quik removal alloy, even though I have some.
3) Then I took apart the Alpha rotary switch, p/n SR1712F-0204-15K0A-N9-N.
Not good. The black plastic actuator has two metal brushes on the bottom that are simply melted into the plastic. One of them had broken off. This means I cannot simply put the metal brush back into a slot or hole. I will have to find a glue that could grip enough on the black plastic in spite of the grease OR re-melt the black plastic perfectly to hold the metal brush. Neither approach will be easy or durable. It also means that I cannot transfer the 15mm knurled black plastic actuator to a new switch body, as it is the part that is actually broken, not the base.
I asked Mouser if they would order the correct part number (15mm knurled shaft) and they recommended the in-stock 20mm flat shaft instead. I guess they don't want to order the correct part for me.
So, right now, I am asking MXR tech support if they will sell me just the rotary switch for a reasonable price. If that doesn't work, I'll ask Alpha Taiwan if they'll sell me just one sample unit. If that doesn't work, I'll order the wrong part from Mouser, cut the shaft from 20mm down to 15mm, then change the knob from one for knurled shafts to one for flat shafts or solid shafts (with set screw).
What a pain! All this due to a tiny piece of melted plastic breaking off.... (see photo).