My 3 year old is also very bright, she will be 4 in march, and have thought about seeing if there is any way the school will let her into kinder early, because she is more then ready know, She is so bright that I sometimes have to remind myself that she is only 3 and other people too, but putting her in a learning center isn't going to give her that leg up, plus she is with someone who loves her and is giving her something more valuable then any school can. look into some extra activities things at the library, or dance classes those kind of things that both girls can do w/ grandma.
Know to the girls constant fighting my oldest 2 are 11 months and 20 days apart ( I never did them that close togather again) it is crazy and frustrating and I didn't know if they were going to survive thier toddlerhood LOL, or me for that matter, one thing I did find that helped was trying to cut out the competition they felt whether it was over a toy, a cup, who has more ectI can't remember how I did that it was 16 years ago, but I can tell you it does get better the older they get, and it was a lot of work early on, but well worth it, mine were a boy and girl, but girls tend to take a lot more work I have 4 boys and 3 girls and my girls still fight nonstop (they are 16,14,&3) but they will defend each other also, girls fight but they are also very close to each other, it just doesn't happen until they are a little older and sometimes not until they become adults, that was the case for me and my sister.