If it splits, nail polish can trap germs,Being in the grooming biz for so may years I will spare you all the stories I can tell about what nail polish can do to a dogs nails.
Here is what you do, Petsmart, petco, what ever, Find a product called Quik Stop it does work for stuff this this minor (small cuts) or if I have nick a nail in cutting I will use, super glue, try and clean the nail the best you can, dry it will take 2 people sometimes get your blood wiped the other puts the glue on and blow on it use a small dryer but get it dry. do that until the blood stops.
I use the stuff for paper cut take the sting away.
And as far as the nail let it grow back natural, just keep the pups wound clean.
Unless You are the lead Dog the View never Changes !