The situation has been resolved, thanks for all the replies!
In the end, we ended up shipping it via DHL. Turns out that you are in fact allowed to ship lithium batteries to Germany, especially if they are small and contained within a device such as an aad.
Only Post Australia won't do it for whatever reason.
But yeah, just don't try taking your rig on a Qatar Airways flight to avoid this situation entirely. I have taken my rig with me on a LATAM Airlines flight without an issue.
Are there any airlines you would recomend or advise against when travelling with your rig?
Maybe we can create some kind of index to prevent this kind of situation in the future. What do you think?
Cheers and blue skies!