Switching between inverting and not inverting a small signal
My goal is to replace the mechanical DPDT switch in this guitar sustainer circuit with something silent, controlled electrically, and can be assembled with through hole components.

There's no such thing.
I've installed hundreds of true bypass systems in guitar pedals over the years - the same applies to your guitar. DPDT switch is the only reliable way. You'd need to install an extra tiny DPDT switch on the guitar.
If you insist on the electronic solution, RG Keen has a series of articles on quiet pedal switching which you can adapt into an active circuit for your guitar. But all solutions will likely involve FETs in a flip flop circuit which will insert a few solid state components in your audio path. This will go against your requirement of it being silent (while it might not go pop when you switch, it will necessarily load your signal). Also JFETs won't be through hole unless you use a breakout board.