This is not an answer unfortunately but I am searching for the same information. A friend is trying to transfer his LR6 licence to me as he finds it too difficult to use. I've used previous versions so I'm happy to receive it but he can't find how to get through the Adobe Wall Of Confusion for a simple case number to begin the transfer. He asked me to help him. I've followed the instructions here How to Transfer an Adobe Software License Between Two People | ProDesignTools and been bamboozled by adobe's never-ending circular instructions too. Seems like you just get dumped in the forum. Disgraceful service. They obviously want to make it extremely difficult to perform a transfer. Can you imagine if they did this to prospective customers wanting to buy products....oh wait, they do, It is very difficult to know how to buy a standalone version of LR6. They try to push you into Creative Cloud at every turn. I was also disgusted that I made an Adobe ID but when I went to use it I was asked to input my date of birth before I could access it. Why not just ask for that upfront when the ID is being created? Sneaky and underhanded at every turn. Did I say I'm disgusted? This has turned into a rant now...back to topic...
The best option may be to just send in the transfer forms without the Case Number, although they advise this may take longer. What a load of ...
To answer Andy-
I have found that the earlier versions of Lightroom would simply work if you had the Serial Key to match the installed version. (If the v4.4 was an upgrade, she will also need the Serial key from v3 or earlier!) and registration was not essential.
If you registered your v4.4 then the Serial Number could be found at you Adobe Accounts page- and you may be able to de-register it there also.
Also- You can look in C:\ProgramData\Adobe\Lightroom and open the .lrreg file (it it exists) with a text editor such as Notepad and retrieve the serial number. You will have to change your file preferences in Windows so that you can see hidden folders in order to get access.