Your GP is unlikely (and hopefully not allowed!) to retrospectively issue a sick-note to cover time off work for something other than being ill.
I get a couple of days but it’s supposed to be for pretty close family, having to use it next week for a funeral. For what I needed last week I used some of my annual leave & the rest i’ll owe them in hours in the future.
It goes on our timesheet as a particular booking number, which I assume gets absorbed by the company as a whole, so perhaps their way of covering you for taking more than is technically permissible in the Ts&Cs?