Thanks for the help! I took another look at snapshots just now and realized that I can change the output block volume to -120dB for the inactive path in each snapshot. That solves the problem because now I can have two totally parallel paths and just snapshot between them, muting whichever one is not active.
I had previously tried snapshots without that little tweak and you could hear the raw guitar sound from the inactive path, even if all blocks were disabled.
Here's a summary for anyone else out there that wants a very easy way to use two DSP-intensive amps and easily switch between them:
1) Put your clean amp on Path 1 and your dirty amp on Path 2 (or vice versa), leaving the paths completely independent from each other (parallel paths).
2) Set both paths to have the input you want (guitar most likely) and output you want (multi most likely).
3) In Snapshot 1, bypass all blocks on Path 2 and set the Path 2 output block volume to -120dB (set up Path 1 however you want your clean sound).
4) In Snapshot 2, bypass all blocks in Path 1 and set the Path 1 output block volume to -120dB (set up Path 2 however you want your dirty sound).
5) I'd recommend changing your footswitches to show snapshots if you're using this scenario, which is a under Global Settings -> Footswitches -> Preset Mode Switches.