I’m not sure how much I can help you because I don’t have Ableton Live.
It’s true that Appendix D of the AF manual only includes the CC and NRPN assignments, but the transport and sync settings are described in the section about the MIDI CONFIG menu.
The following thread seems to indicate that Elektron’s MIDI sync implementation is generally sound:
The official MIDI system real-time messages are Start, Stop, and Continue.
What does Live do when you are stopping and starting it? Are you stopping it, or are you pausing it?