Now the only way I can do this without joining the local flying club (€800 membership fee!) is to fly as student pilot in command with an instructor, regardless of the fact I'm not a student.
Not being a student probably does not matter.
The presence of the Instructor typically trumps your acting as PIC. If the club flying rules state that you not being a member requires the presence of the Instructor and for him therefore to be PIC - then you are stuck.
It is probably the requirement that you become a member before you can fly as PIC yourself in one of their aircraft. Many clubs require membership in order to avail oneself of their services.
That €800 membership fee probably also covers you for hull damage etc which as PIC - you would be responsible for.
Reading LASORS or contacting the Spanish authorities is irrelevant. If the club set rules then you abide by them or go elsewhere. It's their ball.