Stop me if I'm wrong, but it seems misleading to ask whether or not the problem is in P. It seems to me that it is better to ask more generally whether the problem is tractable. There are a lot of games that are solvable in exponential time (not in P) that are still tractable because the branching factor is low enough that simple methods like minimax searches work well.

Take chess as an example. It's an EXP-TIME complete game with a branching factor of 35, and AI techniques win pretty much all of the time. In this case you're looking at a much simpler game (if you treat it as an expectiminimax problem, you alternate between turns with a branching factor of less than or equal to 4 and ones with a branching factor of less than or equal to 32). The game is much less complicated than chess. The game tree is really small, meaning that it should be tractable to most AI techniques.

Even though it is reasonable to 'solve' the problem, there are board setups that are unwinnable. Since the player has a lot more power over the game in placing tiles than the agent does in moving them, it is easier for the player to force the game to one of the unwinnable states than it is for the agent to prevent these things. Even if the agent is acting optimally, the player should still be able to win.