Okay, so it sounds like the destination host might be filtering on the host header value in requests. You could try testing this from the LTM command line using curl to insert a host header:
curl -kv https://1.1.1.1/index.html -H "Host: www.example.com"
If that works, you should be able to use the iRule in my original reply to rewrite the host header to a static string. You'll need to include a server SSL profile on the VIP to have LTM accept clear text HTTP from the client and establish an SSL handshake on the server side.
Aaron