Sorry – I'm running out of ideas. If Tip+Rng works, and EXPTIP is set to SWITCH, that suggests that the switch itself is good, as supported by the fact it is OK with other pedals.
The EXPTIP seems the most likely. Can you explicitly confirm that this is indeed set to SWITCH ?
If it is set to BOTH, this would give the symptoms you describe (except that Tip+Ring would only give you Ring).
So, assuming that you are using V4.8, this should work with the switch, otherwise we would have noticed it.
Otherwise, it would be helpful if you could borrow a different switch and a different stereo jack lead to see if these make a difference.
If all the above fails, we would have to ask you to send in the H9, the switch and the cable, which would be annoying for you.