First, no chemicals are going to lighten the tattoo. If such a thing existed there would be no laser industry. Second, if someone wanted to start working in on a tattoo that's a cover-up of a cover-up after only 4-6 weeks old I'd find someone else.
You say you're not in a hurry, but this coverup isn't even healed and you're looking at monkeying with it again and you are setting a one year time line for finishing your sleeve. Each time you get a tattoo, scar tissue forms as part of the healing process. This will be the FOURTH time you're working that skin.
You don't want to hear it, but that skin needs to really heal and it's going to take time to see how much the tattoo lightens up. If you just jump right into this you'll be back on here in a year asking how to fix the cover-up of the cover-up of the cover-up of the bad tattoo.