Normally, the Unfit letter is generated by the CAA"s medical record system when a pilot"s flight status is changed from 'fit' to 'unfit'
To generate that change an AME needs to have you in their practice, and the pilot gives consent for them to be added to an AME"s practice.
But beyond that it is certainly possible to generate unfit letters but this would involve normally a discussion with the CAA's AMS and maybe some airline's departments with AMEs attached to them may have arrangements with the CAA allowing them to issue unfit letters to their pilots even when they are not 'on their books'
I don't think any AME would generate such a letter though or do a change of flight status without having good cause for it.
Hope that is helpful.