I don't think it will work, because you need to gain speed or loose speed to achive a stable orbit, which means you need a way to affect the moon's speed only one time, and then do not affect it anymore. In third level you can do this by affecting moon's speed once at the start.
A while ago I made a prototype similar to your game, where you need to build you rocket and fly it to the finish while being affected by gravity from planets(kinda like "Bad piggies" and "Angry birds Space" mashup), real life gravity formula do not work well, even KSP uses the "sphere of influence", so cheating with gravity was essential.
Also I can help you with the visuals if you want to release your game(I have some skills in pixel art)