I'm not gonna bother explaining how I did this (my Excel document is glitching out and there are black holes on the page lol) but basically the five Pokemon who learnt the lowest amount of TMs (Wobbuffet, Wynaut, Spewpa, Burmy, Tynamo) all learn their TMs through level-up, so this question became significantly more difficult to answer. But we have reached a conclusion! The Pokemon who can learn the least TM-but-not-level-up moves in Sun & Moon are...
14 moves - Pyukumuku
16 moves - Kartana
17 moves - Trapinch & Snorunt
18 moves - Rockruff
Compare this data to Pokemon who learn the least amount of TMs:
1 move - Wobbuffet, Wynaut & Spewpa
2 moves - Burmy & Tynamo
17 moves - Pyukumuku
19 moves - Eevee
20 moves - Shelmet & Kartana
When only looking at how many TM moves each Pokemon learned, Pyukumuku sat above Burmy and Tynamo as the Pokemon who learnt the fourth-lowest amount of TMs (including Pokemon who didn't learn TMs). Essentially, Pyukumuku retains this spot, thus rendering my efforts useless... but there were some notable changes. All of Eevee's TM moves are exclusively TM moves, thus Eevee was saved from reaching the bottom of the number-of-TM-but-not-level-up moves ranking! Clearly, the asker's modification in criteria changed the rankings up quite a bit.
Okay, I hope I'm done now.
Hope I helped. :)
Source (literally the best thing ever) and Excel