I use a white lacquer stick on my firearms. I bought it from Brownells. I mostly highlight the reciever rear and sight drums on my Garands, M1A's, and AR-15s.
First, I rub the stick on a piece of cardboard to get down to fresh lacquer. Wipe of the edges of the stick to remove crumbs.. Then heavily apply it so that it completely fills the marking. I wipe off with my fingers and palm, only. Clean your hand with soap and water, dry, and rewipe until the surrounding area is clean. On phospated surfaces, the surface may still have a bit of a white appearance, and you will not be able to remove any more with your hand. For me, mostly, this is OK. Once dry (after 24 hours or so), I apply whatever oil I am using at the time....NOT Kroil or anything else that may have a solvent in it.
If I want to remove all the white residue, I use a cue-tip and mineral spirits or spray carb cleaner, and delicately remove it from around the stamping/engraving. I only use a DAMP cue-tip, not enough to make a run or seep into the stamping/engraving.
I tried cleaning the area like you, and found that using ANY type of cloth or solvent removes the lacquer.