Investigation of Subarachnoid Haemorrhage

Had a patient recently who came in with sudden onset occipital headache that sounded like an SAH. Unfortunately we struggled to LP him and I had to contact the anaesthetic team. The anaesthetist asked how the LP would change management as if it was positive we would end up doing CTA/MRA anyway. I was kind of stumped by this and my consultant ended up agreeing with the anaesthetist so we ended up not getting an LP and just getting MRA...

Is anyone able to shed a bit more light on why we bother doing LP if we can just do a CTA/MRA anyway? Does it make that much more difference in ruling out a SAH? How do I rationalise it to someone else?