I've troubleshooted the problem on the Facebook group also and 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 and the first note of the loop is missed.
6. If I choose noSPP in the MIDI output, 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.
https://imgur.com/4ULkjvD7. The note hangs don't happen if I insert a MIDI Note Sanitizer JS plugin in front of a synth plugin.
8. 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 or MIDI file.
So, I sort of got workaround for the original issue, but it's still broken for some reason.