This is not an option when making a Sentinel Feat Character, nor are the rules for it stated anywhere on Wizards of the Coast’s site or in supplements. It sounds like a Rogue class feature (specifically Sneak Attack). You might want to contact their support team and ask about this particular feat before starting your campaign; this way you’ll know players don’t have to dumb down their characters because of something their DM added, which could be very frustrating for them.
No smite tyrant ability states that they deal damage as opportunity attacks so I can’t imagine why it would work with Sentinel thinkin of hands on combat without use evasion against monsters that come inside melee range. Sentinal is mostly about being first in line to take on whatever comes your way. When opportunity attacks are missed which seems to happen quite often with a player that was not aware of it or use his action to tell the DM that he does not want to do it then the attack misses which is a good thing for you as Sentinal Feat can be broken very easily if there would be no miss chance with them and that is something that we don’t want.
The Sentinel feat doesn’t provide an opportunity attack. A monster, such as the tyrant described in this thread, can still make an opportunity attack against a character with Sentinel if it leaves their threatened area (which it does, by using the Throwing Axe on its turn). This will almost certainly hit the character, but if it does so it will not trigger an attack of opportunity because that would be between the monster’s two attacks.