Although we have:

B:"Remove Recipe Book Button"=false

The icon is not showing up for players on our 1.19.2 forge server; I'm not sure what the cause for this is, as it seems to be working in single-player and LAN hosting.

Clients also have the setting matched to the server.

Installed FW version is 1.19.2-7.1.3

You might have other mods that include this feature are you playing a custom pack? or something already on curseforge?

I'm using a custom pack I have put together and which is available on Curse (Nermagitek is the pack name)

You should be able to see the modlist on this page for the current modpack release:
https://legacy.curseforge.com/minecraft/modpacks/nermagitek/files/4590389

Is there any way I can log something that might indicate why the book icon isn't there even with the setting correctly set to false both on clients and the server?

If the setting is false on the client, FB will not remove the button https://github.com/Shadows-of-Fire/FastWorkbench/blob/1.19/src/main/java/shadows/fastbench/ClientHandler.java#L26

The setting does nothing on the server.

However, there are other mods known to remove it as well. Unfortunately that list isn't known to me.