Thanks Nikolaos, yes it seems the best way to achieve this would be via a LCS project.
I had issues before creating such a project, since my company did not have access to all the pieces, but that has been solved now.
So, I created a AX 2012 LCS project, and hoped I could find the download for upgrading. However, the only thing I find in LCS is the Update tile which allows me to download an application that I should run on the AX machine. This download is the same one I have found before (open on Microsoft pages), and the first thing this application wants to do, is to either a) connect to LCS, or b) select a already downloaded package.
I tried to select a), but then it complained about me not being connected to internet. I dissambled the update application and found that it was trying to use a TLS version that was not applicable using that version of dot net that the update app was built on. Fiddling around with installing newer versions of dotnet, and installing some well selected Windows KB's, I was able to get around this.
Now, the update application launches a new windows with an IE control inside it, that asks me to login to LCS. But, when I do so (using my corporate Azure AD account), the Microsoft login pages simply says that it could not log me in (but also states that I'm logged in ?!). The same thing happens if I launch IE manually and try to login to LCS. I installed Chrome to check, but there it works fine to login.
I tried safe-listing all sites that are involved in the login process and also lowered the Internet Explorer security down to the lowest I could figure out, but with no luck...
Btw, as a side note I also tried doing a Windows Update on the machine. It says it was searching for updates, and I let it run for 2 days, but it did not complete so I had to shut it down.
Option b) would be nice (installing a downloaded package), but I cannot find such an option anywhere in LCS.
Does anyone know how to get a hold of such a package? (for example, AX 2012 R3 CU13).