Badge Arcade app uses SpotPass to get its data.
If I recall correctly, it usually pulls two encrypted containers (data_v130.dat and data_v131.dat) that hold stuff for current week, like cranes, badges, text strings, etc.
Sadly, I don't think that anyone ever bothered backing up those files for every weekly update, but it's likely that the only actually unique stuff in those is dialogs with rabbit.
There's another pair of files (allbadge_v130.dat and allbadge_v131.dat) which, like the name implies, contains all badges, sets, crane backgrounds and icons, minus some badge and crane data that is already inside the app + latest patch combo.
You might want to look into this tutorial:
https://gbatemp.net/threads/get-any-official-badges-without-other-users-data.476300/
It uses those "allbadge" files mentioned above, and the tool pack from there should give you an idea where the SpotPass data is downloaded from and how to decrypt it.
So, even if weekly archives are not preserved, it might still be theoretically possible to craft custom ones with any cranes and badges ever released, as all of the most important data should be available one way or another.