Ok so if I have the chain right we have:
Ableton providing MIDI notes and tempo via USB to Beat Step Pro (this can be a two way connection to Ableton unless preferences are set otherwise)
BeatStep Pro Sequence 3 (purple) via MIDI to the drum hardware (one way connection away from Ableton assumed)
Using MIDI Thru to pass Beat Step Pro MIDI (or drum hardware MIDI) on to Hardware Synth (one way connection away from Ableton assumed)
I assume you have audio out of drum hardware and synth going back to Ableton or off to a mixer or monitor speakers. But even if you were sending audio back to Ableton it would not fire MIDI tracks in Ableton Session view.
The only thing in this chain that can fire Ableton clips is the BeatStep Pro via the 2 way USB connection.
The two way USB connection carrying MIDI to / from Ableton is a preference that can be changed between: both ways, out only or in only, plus track, sync and remote control are sub groups of this also.
In Ableton preferences, your one way chain would probably succeed using the following.
MIDI ports in - Track OFF / Sync OFF / Remote OFF
MIDI ports out - Track ON / Sync ON / Remote OFF
If you are not intending to record MIDI from the BeatStep Pro to Ableton then the MIDI Track In track should be OFF.
If you are not intending to drive Ableton with the tempo from the BeatStep Pro then Input sync should be set to OFF
If you are not intending to control Ableton controls with the BeatStep Pro then the input remote needs to be OFF.
If you intend to send MIDI from Ableton to the BeatStep Pro then Output Track should be ON.
If you are using Ableton to provide the tempo signal to the BeatStep Pro, then the out Sync needs to be set to ON
If you are not intending to control the BeatStep Pro controls using Ableton then the Output Remote should be OFF
Also, the BeatStep Pro appears twice in Ableton MIDI preferences, the first one is the one you modify track / sync & remote. The second one you don't touch or use. It is a background channel needed by the BeatStep Pro to work properly. All Arturia devices have this pair of detected inputs, their literature say to leave the second one alone on default settings.
I did this from memory, with only a cursory look at Ableton during, so apologies if it is not entirely correct, but I tried to be accurate.