R..
I wouldn't use peroxide on it... the bottle I have says not to use on children under 2 years of age.. Plus, on a post I made asking what mothers use to clean sores one mother pointed out that it takes the ability to heal away from the skin... I looked it up and it's true.
I wouldn't worry about it too much as long as it shows signs of healing. Now, if in a couple of days it is still 'fresh' looking I would get it checked out... or if it scabs over and that keeps tearing open, then stitches may be in order. Other than that, maybe find some mouthwash that has no alcohol and apply it to the cut with a cotton ball to keep the inside clean, and a bit of neosporin for the outside.
I have had 9 brothers... PLENTY of split lips in our family!!! LOL! The only one that went to the ER was the one who nearly ripped his lip OFF (Long story there...) We have had a few instances of teeth going completely through the lip, and they always healed up just fine.