Ok first and for most this will not work if you have a banned console. What you need 2 Xbox's 1= Stock 2= Modded 1 Memory Unit Process 1.Sign into live with your stock console recover your gamertag to the memory unit 2.Put the memory unit into the modded console and sign in (not to live!!!) play games get achievements. 3.When you want to play on live again move memory unit back to the stock console and sighn in to live gamerscore will be updated on live and you can still play original games. 4.I suggest that you only use the memory card for your profile and not game saves or anything else when transferring from console to console. Its not the greatest solution in the world but it does work for people who are worried about playing backups and not getting credit for the achievements they unlock.
in case any data gets logged on to the hd with a bad rip it would than be shared with the stock console
that may be not 100% sure on how they do the ban flags. but i thought id suggest a possibly safer route. If there is such a thing lol
its cool. but i really doubt the HD has anything to do with it. otherwise everyone that has used their original HD on a new console would have been banned a long long time ago.