Offizieller Kommentar
Looks like those holidays were a lot longer than expected, and you are super patient!
Here's what they had replied a while back, but I missed the ticket response:
Currently, there is no way to record Beat FX on iOS.
It can be recorded on a computer connected to the DJM using a USB-B port.
You can use REC function of rekordbox.
The mixer REC OUT audio is sent to REC CH on rekordbox as the default setting.
Thus, you can record the output audio from the mixer as it is.