Loctite is basically like glue. Red is a lot tougher than blue. Red is usually used for a permanent hold. The blue may work better for you if you plan on replacing your rear sight in the future. I've never heard of green loctite, but it may work great for your application.
Since you can move your rear sight by hand, I would slide your sight to one side, apply some loctite inm the beaver tail, then slide the sight the opposite direction and apply some loctite to the other side of the beaver tail. You need to be sure that the areas you apply the loctite is clean and free of oil. Also, you want to be sure that your sight is where you want to to be in the beaver tail when you finish, clean up any excess loctite, then let it set according to the manufacturer's instructions. Blue loctite will not damage your P-120 in anyway, so I wouldn't be afraid to use it and you'll still be able to replace you rear sight in the future if you like. Good luck.