That was actually pretty easy to setup.
I installed SyncThing on my Windows PC using https://github.com/Bill-Stewart/SyncthingWindowsSetup and set it up to monitor a folder which contains the non-Steam games I want to sync to my SteamDeck.
On the SteamDeck, in Desktop mode, I installed the SyncThingy app via the package manager and followed the instructions to set it up as a service that starts at boot time, so it will even work in Gaming mode.
Once that was running, I went through the process in SyncThing to synchronize my PC and the Steam Deck, which does take a few clicks and confirmations on both the PC and the Steam Deck, but after that it just started copying the game folders automatically.
After a game had sync'd to the Steam Deck, I added it to Steam, switched back to Gaming mode and played it for a bit. After saving my game, I checked on the PC to see that the save files that added to the game folder on my Steam Deck also now showed up on my PC.
The settings got majorly reorganized so I suspect they got reset in the process. Perils of TestFlight!