Tim,
MAC.OSX.Trojan.FakeAlert is one name that some security companies use to refer to MacDefender, which is very old malware that has long been extinct. See the following examples on VirusTotal, which some vendors call MAC.OSX.Trojan.FakeAlert, but others call MacDefender.
https://www.virustotal.com/en/file/...666ef2dcb337ce9fa6cd653c6d2903cef25/analysis/
https://www.virustotal.com/en/file/...d82cf88733dc4f397c955b1ec8d5f40cde9/analysis/
For some additional information about MacDefender, see:
http://www.thesafemac.com/?s=macdefender
Now, it seems rather unlikely that your backup drive could have a copy of MacDefender in it, unless it's got data in it that is several years old and you once had a copy of MacDefender on your hard drive. Further, it's utterly impossible for any threat to remain on a hard drive after it has been erased (assuming that you erased it using Disk Utility). So, as I see it, there are two possibilities.
First, it could be a false positive that's triggering on something specific about the Time Machine backup. That would explain why Bitdefender is only finding it on the backup, and why it came back after erasing the drive.
Second, it could be that Bitdefender is using that name to refer to something that is not MacDefender. (Why they would do such a thing, I don't know... but naming conventions of malware are quite inconsistent.) That could explain how it got back into the backup again. However, I can't say why it would be finding it only in the backup and not on the main hard drive; that doesn't make much sense. For that reason, my money's on the first explanation.