Please Open Lightroom
Go to Lightroom Preferences
Click on the Lightroom Mobile tab
Click on Delete All Data
By following the above steps it will sign you out from the Lightroom Mobile from the Desktop App, also it will delete the sync data to Lightroom Cloud.
Let us know if that helps.
I have this same problem with the same greyed out button - was there ever any answer to stopping it from asking me every-time I start the program to sign into Lightroom Mobile?