You could use John Dvorak's trick--buy all your Amazon items through an affiliate link for an affiliate you own. If you purchase enough, your affiliate rewards will cover your Amazon Prime membership.
I tried this a long time ago -- maybe 10-12 years back, not long after they first introduced the affiliate program -- and got a friendly but pointed letter from Amazon to knock it off. Have they changed the rules?
It's the same Amazon account for Associates and purchases... It's quite simple to match up. You may able to sneak around with different email and mailing addresses, but you still have cookies to deal with.