Fangs of the Fire Snake does not couple with Tavern Brawler feat.
The description of grappling says (emphasis mine):
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. [...] The target of your grapple must be no more than one size larger than you and must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check instead of an attack roll: a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use).
You must meet 3 requirements for grapple a creature that is 15 ft away from you:
The target must be no more than one size larger than you: this depends on the situation and on the enemy you are facing, this is independent on the interaction with Tavern Brawler feat.
The target must be within your reach. Fangs of the Fire Snake increase your reach by 10 ft, leading to have a 15 ft reach: this requirement is hence met.
You must use at least one free hand: the Fangs of Fire Snake creates tendrils of flame erupting from your hands and feet and let you attack with them, not with your hands/feet.
The last requirement is not met, since Fangs of the Fire Snake allows you attack an enemy at a distance of 15 ft with tendrils and tongue of fires, not with your hands: thus you can not grapple a distant enemy as a bonus action unsing the Tavern Brawler feat.