AASA buys their insurance from the same broker in London as CAMS, AMSAG (and presumably MA).
The cost of adequate insurance is hard to avoid, regardless of who you get it through.
In the case of AASA vs CAMS, CAMS is a bloated organisation* with far too many suckling on its teat, and is ridiculously risk-adverse for a motorsport sanctioning body, and an obsession with control. And poor customer service (although they've gotten a lot better in recent times).
For these reasons, it is easy for AASA to offer a superior product to CAMS.
I know lots of people love to hate MA, but MA's sins are far less - presumably this would make MA's product closer to AASA's?
I've had a bit to do with AASA, and have no hesitation in saying that they would be worth a phone call. They tend to say "tell us what you want to do, and we'll tell you how much it will cost", which freaked out those who were used to the CAMS style of "we will tell you what you're allowed to do, and how much it will cost".
*As demonstrated by their incredible ratio of staff:turn-over.