Is there anything he can do about or with the dementor other than destroy it, which is far too obvious?

Go in front of it and let the horrible personality possess him again. After making some sort of precommitment, I suppose. This solution was foreshadowed in the omake "Lord of the rationality".

But in truth, I lost faith in Eliezer's ability to come up with realistic solutions when Harry miraculously survived the first dementor attack in Azkaban. He will probably just have Harry use his main character powers again.