This would be cool and is hard hard hard. I speculate PhD thesis-worthy hard, or harder.
The reason is the sheer amount of variety in Dominion, and somehow the learning AI should be able to smoothly integrate it all. As DStu points out, there are many kinds of games: combos, slogs, engines, etc. Tactics and strategy are both important, and fairly different. There is randomness, so brute-force search can't give the AI much help, yet tactics and endgame play rely on precise calculation. There is hidden information, since you don't know what's in your opponent's hand.
In addition to that, the magnitude of the task is huge because there are so many kingdom cards. Each one is going to need some degree of special-casing and many present unique challenges. For example, games dominated by Possession are dramatically different than normal games.
Edit: That all said, I'm pretty sure it's possible with today's technology to create a Dominion AI that outplays top humans. But it would be a TON of work and cleverness. It might be best to attempt it with a team instead of just one person.
Ultimately, I think the main barrier is that there simply isn't enough reward for the work required. In the case of games like Chess, Go, Poker, etc., that have received and/or are receiving research effort, they have been around for hundreds of years, are very popular, and have rules than are simpler than Dominion's.