I've troubleshooted the problem also on the Facebook group and Arturia Beatstep pro forum.
So far no idea what the problem is, but here are the latest findings.
1. I tested it on another computer - a 2011 Macbook Pro - the exact same result.
2. Then I did a reset on the Beatstep pro - same as before.
3. If I disable the control messages on the input, the skip/stutter problem goes away, but then the playback buttons obviously don't work.
4. I also get frequent MIDI note hangs with the Beatstep sync set to USB, otherwise not.
5. If I choose a specific channel for the input control messages, then the stutter resolves, tempo is in sync, but transport controls don't work.
6. If I choose noSPP in the MIDI input, then I don't get No skip/stutter, but the DAW play/stop control becomes glitchy - edit cursor skips forward with each play/stop and needs to be clicked very shortly to start playback.
The short recording problem still persists - now even shorter - the recording stops immediately after I press record and get prompted to save the microscopic wave file.
https://imgur.com/4ULkjvD
(Image code doesn't work for imgur image for some reason)
So, I sort of got a couple of partial workarounds for the original issue, but something's still broken.
For the time being using the MIDI control input with a specified channel and no transport control from the Beatstep pro.