After installing Izotope Sutter Edit, the VST3 version doesn't show up in the Synth list under in the FX bin
but the VST2 version does. Both are configured as a synth in the plug-in properties and show up fine as FX
after running the plugin scan.

IIRC, both the VST2 and VST3 originally scan as FX plug-ins. Starting with X3 configuring FX plug-ins as synths is not necessary and I do not do it. Instead, I use the VST drop down in the standard header and enable the appropriate MIDI settings for the plug-in. That said, the problem changing the plug-in options on VST3 plug-ins has been reported.

Configuring the VST3 version as a synth or not still doesn't show up in the synth list.
Not for Stutter Edit anyway. It's not a big deal as long as the VST2 version works.
Just thought I'd point it out.
 

Which "appropriate MIDI settings"? I don't see anything that says "enable MIDI in" for example.

The ones that were staring me in the face, you mean? :-)