For that type of injury, topically is the way to go. Get a container if you can find one that's just a bit larger than your finger.
Whenever you don't have time that allows you to relax like watching TV, or any thing that doesn't require two hands soak the finger in CS. Or buy a small spray bottle and spray the finger whenever possible.
The more times you do either way, the better.
CS will kill both the good, and bad bacteria in the gut.