BLMs actually don't spend much time in Umbral Ice, let alone III. Please look up BLM's rotation thread.
TP regen is the auto rate you gain back TP, not through skills. Invigorate helps but that doesn't count towards TP regen. Also MNK's optimal Invigorate use is at 540 TP so even if there's a TP tick after you use it, it's not wasted. PLD has Shield Swipe which is low TP cost, high potency as a mean to help with TP. WAR has Berserk though the use of both skills are limited.