4 AA batteries will have a much larger current capacity than a 9v pp3. A pp9 might match.
You could put a large cap in parallel with the battery to meet short term loads and lower source impedance before the reg.
try 1000uF 16v electrolytic.
I don't know your current draw but 4 AA will last a lot longer than a pp3 and cost less to maintain. 4 C cell nicads would be even cheaper to maintain.