Aspirin definitely interferes with clotting and increases bleeding during and after surgery in general. In most delicate surgeries, when bleeding is excessive, it may be difficult or impossible to dissect accurately and safely. Excessive bleeding in the postoperative period brings on problems of its own, including the possible need for drainage of accumulated blood, reexploration of bleeding areas, and excessive scar formation. I would definitely postpone a cosmetic procedure if you were my patient.
With regards to spine surgery in particular, there are many types, and differing degrees of urgency - bleeding may be a smaller risk compared to 2 additional weeks of spinal cord compression, for instance. The cosmetic/plastic surgery specialists on this board are probably ill-qualified to give you specific advice regarding aspirin's safety. Your surgeon would be in much better position to advise you.