That wouldn't work without more changes in his kit.
See, Kayle does almost exactly that. But Kayle has sustain, a targeted slow and her ult to survive enemies all-inning her. All Teemo does have are shrooms, which can get disabled, and blind, that doesn't do shit against many champions.
If Teemo had timer on his Toxic Shot, he could never ever use it against say Renek or Pantheon. Because the moment it's put on cooldown Teemo gets completely zoned from all CS until it's back up - they can just jump on him and take 70% or more of his HP free of any meaningful retaliation.
If Teemo had CD on Toxic Shot, he would also need a ranged slow like Kayle does - so that when he uses it, he can reliably hurt his enemies. I don't think we want him being even more annoying than he is now.
Simply increasing his damage for exchange of CD also won't work. 50 on-hit damage plus 30 DPS from poison without items is already very high, if it was any higher, Teemo who got a bit ahead in levels and items could straight up melee form many threats, and still win. Notice how Kayle's single target damage output is actually a bit lower than Teemo's, even tho it comes with CD. Instead she gets AOE, but Teemo won't get that or he would be too similar to her.