Why won't the doctor put me out for a lumbar puncture?

Risks . Putting someone out (ie. Anesthetising them) carries some risk. Lumbar punctures are usually well tolerated with just using some local anesthetic (numbing medicine).
Generally not requir. This is not typically necessary. Perhaps you could ask for IV sedation prior to the procedure if excessively anxious. Furthermore, general anesthesia may cause vasodilation which is undesireable for an lp.