If you're matching 3 enemy tiles (as it is implemented now), the AI is some overarching system that looks for an enemy match. I don't think this is good.
If you implement the enemies as a modifier on regular color tiles, you can lessen the amount of enemies on the board; it's realistic to implement specific AI for each of these enemies and not have every single misstep become a battle.
When you don't complete a match (or something to that degree), allow the AIs on the board to make a single move.
A good portion of making the dungeon "a bejeweled mechanic" is to keep the game from feeling like "a dungeon crawler with a dumb bejeweled gimmick minigame". I am less inclined to think that "avoiding matches" is a good idea; I do like the idea of dropping a staircase to the bottom of the board to enter the next level. Ultimately I think you want something that differentiates the bejeweled playing in the battle and the dungeon screen.
I am still in favor of the "spellbook/combo system" for the battle; perhaps the interesting bits from the dungeon screen come from the AI/powers implemented by the monsters (i.e. one that attempts to get the closest match, one that attempts to run under the staircase, one that modifies tiles on the board).