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Here's the switching schematic. It only shows the power and relay coil sections. I also listed the Mouser part number of the relays. The schematic shows a three mode switching scheme. In my case it is Clean channel to Gain channel, then Gain mode one to gain mode two. For a simple two channel switching scheme leave out one of the relays.

Normally I use a combination of basic switching and grounding techniques. I'm usually doing switching with an amp that uses one input to the PI. So in the case of the sIII you could just use a SPDT relay, instead of the DPDT and just use grounding techniques. But I'd go ahead and get the DPDT anyway, just in case you need a little more control.

I generally ground the gain channel after the first or second gain stage. It all depends on the topology; in an amp that shares the first gain stage, I do the gain channel after the second gain stage. In the sIII, I'd just do it after the TMB's first stage. If that causes a pop then try moving it to the TMB master volume pot's input.

On the normal channel I'd ground from the volume control input.

The closer to the PI you get the less pop effect you're likely to get.

I'd also take one of the Normal channel inputs and wire it to feed both inputs, or add a switch. You could share the first gain stage but this would change the character of the amp.