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Should I Change the design of the (A) site? : Short term answer is yes (more instant sales) and long term is NO (MSN SERP position/traffic lost).
Should I use the exit traffic: Be abusive to visitors with site (B) popup/popunder on exit, No I don't like it...
Should I put a banner for site (B) on site (A)?
Other ways to redirect traffic is very welcome!
Why not just redirect all visitors on domainA.com to domainB.com using a 301-permanent redirect?
The smart search engine 'bots will follow this kind of redirect and combine the PR/linkpop of siteA with that of siteB. The dumb ones and human-edited directories won't do this, and you might want to ask them to update your listings, but visitors will still be redirected to siteB when they click through on an old link in one of these directories.
In order to avoid losing the referrer information, you may want to attach it as a query string when you do the redirect from siteA. If this is important, it can be done.
Depending on what kind of server siteA runs on, you can do the redirect in several ways. Posting your server type may get you more specific replies.
HTH,
Jim