I don't think it's likely that your Alpha Client would upgrade to the Beta Client, it would be a lot of work on Bungie's end to first determine if you already had the Alpha Client on your machine or not and then either give you a patcher which selectively upgrades your client to the beta version (which is a lot of work in and of itself) or the full download in the latter case. Easier to just force everyone to download the darn thing :)