Umm, I think the summary pretty much explains itself. I only tried this with TNO. His level didn't appear to go up each time I clicked the 'level up' button but his max HP increased.
After switching to Morte's stats screen and back again TNO's level had raised to the correct value for the number of times I'd clicked on level up. The 'level up button' was also disabled as it should be and the next level XP value updated to the correct value for TNO's new level.
I have only tested levelling up with with PST.