I would say you probably want to stay in Monarchy until you can either destroy or make peace with Japan.
War weariness is still one of the least understood concepts in the game, IMHO, and I can't say (nor do I think others could with any conviction) how your citizens would react to a democratic form of government. I assume you're not playing a religious civ (since you say that the anarchy would last 7 turns probably), and the thought of going through that long anarchy only to find that war weariness throws your democracy into anarchy right away would be exceptionally painful. A few extra turns in Monarchy won't cripple your research, and when you do come out of the wars and move to a democracy (consider republic!) watch what happens to your tech race with Egypt!
I'd hang tight, try to make peace / destroy Japan, and not worry too much about Cleo unless you want the Egyptian war as an opportunity to expand across the oceans. Then make peace with Cleo and immediately thereafter revolt and go to democracy (or republic!).
Catt