Bartender Blog has this:
Bartender 4 and macOS Sonoma Beta
12 July 2023
I am aware of the deprecated API warning macOS Sonoma is showing for Bartender, and do not need informing about it.
The API this refers to was just deprecated in macOS Sonoma, and the replacement Apple recommend does not yet have the features needed by Bartender 4, so I cannot yet update Bartender 4 to this.
You will also notice a new menu bar item in macOS Sonoma, informing you that Bartender is capturing your screen whenever it gets an image of your menu bar items, again this is new in macOS Sonoma, and currently unavoidable, because of the same API issue. I am working on Bartender 5 right now that will have the ability to avoid this happening.
I will release test builds/betas as soon as it is appropriate.