Oh how I hated cutting out the dairy. I feel your pain! The doctor wants you to cut dairy because your baby is probably sensitive to milk protein. Drinking lactose free milk would only help YOU if you were lactose intolerant. It won't help the baby because,
1. She is drinking your breastmilk and can probably tolerate the lactace that is present in your milk without problem, and if not then your doc would have put her on some sort of hypollergenic formula or drops that you can give to your baby while breasfeeding.
2. The milk protein that is in the lactose free milk will still make its way into your breastmilk.
I had to do a lot of research too to figure this out. When I first started my dairy elimination diet. I was baffled when, after cutting dairy and he was remarkably better, he started getting sick again. It finally occured to me that the splash of non dairy creamer I had in my decaf was the culprit. It just didn't cross my mind that non dairy stuff was a BIG no-no too. :-(
The good news is that you don't have to be dairy free forever. I tried dairy again when he turned 6 months and he was just fine. YAY! Great for me because I am still nursing him. :-D
Here are a couple good links:
http://www.drgreene.com/qa/lactose-free-milk
http://www.kellymom.com/babyconcerns/food-sensitivity.html
ETA:
Rice milk and almond milk are worth a try. I really liked the rice milk, and I still on occasion, will grab some chocolate rice milk for a treat. They even have rice milk ice cream!!
AND an FYI my doc also told me to give up soy and peanuts too because babies can occasionally show an intolerance to those as well. SO no soy milk for me either. :-(