I don't know if you can rebuild the sill cock without removing it or not. Repairing these is not worth the effort in my opinion. I do understand your reluctance to mess up the finished basement, but that is what I would do. After the unit was replaced, I'd build an access door and trim it so that it would not be unattractive. Replace does require access to where the sill cock attaches to the water supply line. Do not even consider it otherwise. You will need two wrenches. One wrench to unscrew the sill cock and one wrench to apply counter force to avoid twisting the supply line. Once the old sill cock is unscrewed, it is pulled to the outside. Usually there are a couple of screws that on the outside that hold the sill cock snuggly against the side of the house and these must be removed. Installing the new sill cock is the reverse of removal. You will notice there is an arrow or similar mark on the inside end of the sill cock to indicate which side is up. Do you need a plumber? Only if you are completely inept with basic tools.