Give the Weyhill Fair a ring, but as they have just had a beer festival there this weekend you may want to give them a day to recover! They have a field at the back of the pub which people camp on. http://www.weyhillfair.com/ It's a great real ale pub.
It is a proper campsite but also has a pub: http://www.wykedown.co.uk/campsite/campsite.htm I haven't stayed there as I used to live in Andover - and the Weyhill Fair was our local. It was a few years ago which is perhaps why I had forgotten the WF only allows camping when they are doing things like beer festivals.
I'm running out of ideas but there are two car parks in Mere, a bit further down the A303 which are free and allow overnight parking. Alternatively I am sure you could stay overnight at Amesbury services as you can at motorway services but of course you have to pay.