I've since replaced the computer, but after downloading the installer for Anthology X and Tverb, it (expectedly) asks me for my serial number.

I'm a bit confused by this.  The installers for Anthology X and Tverb shouldn't ask for your serial number.  To get the license for these plugins deposited to your iLok account, you enter the serial numbers when registering them on our website, but after that, the licenses are in your iLok account and you should never need to use the serial numbers again.  Could you perhaps post of screenshot of where you are seeing this?