Currently the favoriting functionality doesn't work properly with the inventory sorter mod. When using the hotkey or the Gui button, the item will be moved and the favorite icon will remain in that slot. The icon remains there in that slot until the player opens and closes their inventory. Additionally, the item in question is no longer registered as a favored item.

I looked at the source code of Inventory-Sorter mod and found out that it sorts the items on the server side and then synchronizes the slots to the client side. In this situation, I'm not sure how to track the location of items.

You can use client-side mods that implement item sorting functionality, such as Inventory Profiles Next or Mouse Wheelie with Server accelerated sorting disabled. These mods simulate player actions to implement sorting functions, allowing item locations to be tracked. Please download the latest version, which fixes a compatibility issue with these mods.