Am I understanding that you use CLP as both cleaner and lubricant? If so, the brake cleaner will take the place of the cleaner, but not the lubricant. If you use brake cleaner, you WILL have to lubricate with something. Whether it’s CLP or another oil. I keep several different oils in in needle bottles on my bench. RemOil for things I don’t particularly care about the oil staying on (it’s basically just 3in1 oil), Hoppe’s regular oil (I don’t recall the name, but it’s the stuff in the orange bottle), and Hoppe’s weatherguard (mostly for the exterior of firearms that may be subject to the elements. I also keep Hoppe’s no. 9 in a needle oiler. When you make your living on a gun bench, you find anyway you can to mitigate cost. The bottles allow for precision application while preventing me from over using product. I very rarely use a CLP, at all. It usually comes in an aerosol can. I stay away from anything that says “Shake well.” if it separates in the can, IT WILL separate in the gun. Hope this helps.
Oh, and be careful using brake cleaner around wooden stocks. Some wood finishes don’t hold up well to brake cleaner.