Once a week aspirin should be OK as long as you do nothave GI symptoms. While it will affect your platelets and make bleeding more brisk, it will not necessarily cause pouchitis. But, all bets are off if you begin to have symptoms.
All pouches have some level of constant inflammation, so what is worth treating depends on the degree of inflammation. You were placed on a version of mesalamine, which is for IBD, so I would suspect your inflammation is not the minor, subclinical type.
Still, if you always have active pouchitis, it could be that the aspirin is contributing to it. There are many other treatments for headaches besides aspirin and acetominiphen. If you are having headaches as often as weekly, you probably need preventative treatment. Most of those drugs are not NSAIDs or pain relievers, but alter the brain circulation, among other things. I would see your primary doc about options in this area, rather than continuing on aspirin. Another thing to think about is the fact that when you take pain medication for headaches as often as weekly, you are more prone to have rebound headaches, so it is a vicious cycle. Prevention is better than treatment.
Jan