This type of use is quite specific, and not something we had in mind at the early design stage of the Sylphyo/Link. There is no data sent from the Sylphyo when changing base Key (neither any controller setting as breath setting or other control setting, all you can have on the Link side is the result of these settings).
Actually the Link is just currently replicating the Sylphyo internal Core. Wireless transmission goes from the Sylphyo to the Link and not backward (except for radio internal protocol acknowledgment message). Getting data back to the Sylphyo from the Link would be theoretically possible but this would also use radio bandwidth, make the radio protocol more complex and thus increase latency for realtime playing messages.
Changing anything on the Link without providing back information to the Sylphyo would probably not be so comfortable as you would have no feedback about the Link setup compared to the Sylphyo setup. So the most consistent way to make things work was to have the Link replicating the Sylphyo setup and status.
When the Sylphyo Core transmission code was written we had to optimize the data transmitted to get the best latency and radio range possible.
Program change message were added later. Base key was not an obvious MIDI message to be transmitted (and we would still have to figure out how to package the information in the MIDI flow).
Best latency optimized result should be possible when using wired MIDI rather than wireless to manipulate MIDI data.
Thanks for this use case description though as this could help us to think about a solution at some point.
@Paul-Flute did you setup any control to send "'General Purpose 7" MIDI CC in your Sylphyo ?