Hey !
You are all right about the source of the issue.
In Ableton it is pretty simple, simply double click on your MIDI clip and on the bottom of the screen, make sure nothing is written after "Bank" "Sub" and "Pgm".
If you find any numbers in there, you can remove them by holding and dragging down with your mouse. Otherwise, it will send a program change each time you play the clip.
I attached a screenshot to show which part I am talking about.
If this part is not displayed at first when you double click on the Midi Clip, then click on the tiny little notes icon just next to the "E" letter.
Hope it helps