It's possible that the LMF browning solution will bite thru the bluing but I think it will be kinda iffy.
The existing bluing has absorbed oil which really needs to be removed so just go ahead and remove it before you start.
If the existing bluing is degreased, just about any acid will remove it so you can use vinegar or lemon juice. Something made for removing rust like Navel Jelly will work and of course Birchwood Casey Blue & Rust Remover both are good.
It's been my experience that most browning agents including LMF's work best on surfaces that are slightly rough. If the surface is too smooth, they have difficulty "biting" into the steel to start the rusting process.
Due to this I do not recommend using a sandpaper finer than 220 grit.
A couple of sheets of the black "wet/dry" silicone carbide paper is made for use on steel parts and it is fairly inexpensive.
Buy some 180 or 220 grit.
If it were mine, I would just sand the existing bluing off. That way I know the surface is raw steel and it is ready for the browning solution.
Remember, LMF and the other similar browning agents need to work in a humid environment. A steamy bathroom is great.
A non steamy room in the house is iffy to say the least.