I've seen a couple posts scattered about with the same issue I'm having, but none that seem to have resolved (Re: Why can't I install Flash Builder 4.6 in my Creative Cloud subscription?). I have a Creative Cloud membership, but want to use Flash Builder's design tab which is only in version 4.6. The CC app only loads v4.7, so I manually installed the older version. Right now that's working fine on trial mode, but the trial will run out and i have no serial number. My question is, is there a way to get the software activated thanks to my CC credentials, or get a serial number out of Creative Cloud?
So far, I haven't been able to get a response from support regarding if it's even going to be possible to get a serial number. Flash Builder seems to be in some sort of bizarre support limbo where the main tech support staff don't support it, but the people who do don't check tickets or do live chats. Calling the support line gets me redirected to yet another line, on which I've failed multiple times to get a human to talk to.
Here's the update for those who come here with the same issue in the future. I was finally able to get someone on the phone to escalate beyond tier one support (pro tip: social media is your friend here). The definitive answer is that as a matter of policy Creative Cloud will only grant access to Flash Builder 4.7 and beyond. A feature request has been logged to reinstate the design view tool, but as of now there are no indications that it's on the way.