The issue is probably a combination of taste and delivery (bottle/cup). Both of my sons are allergic to dairy. They both took bottles of breastmilk relatively well though, so that part wasn't an issue for us. Are you able to pump at all? Will she take breastmilk in the sippy cup?
My youngest won't switch from the bottle to sippy cup for formula. He'll drink water from a sippy cup, but like your daughter, will only take one or two sips of formula from a cup. So we are still using a bottle. :(
It could also be the taste of the formula. The formulas that are fully hydrolized taste the worst. I don't remember if Elecare is one of those or not. Our allergist recommended that we try Good Start formula for our youngest. It's partially hydrolized (just the whey), so doesn't taste as bad. My son does not react to it (although he does react to whole milk, cheese and yogurt.) But talk to your doctor before you try it.
I would NOT try Almond milk as others have suggested. Our allergist recommended not trying any nuts until at least 3 yo. Rice milk is a good source of calcium, but has no protein or fat, so is not equivalent to whole milk nutritionally. You'd have to make sure you make up for that in her diet.
Also, you could try goat or sheep's milk products, but the chances are she'll react to them to. (Studies have shown that 90% of kids allergic to cows milk are also allergic to goat and sheep milk.)
Have you heard of POFAK? It's an on-line support group for Parents of Food Allergic Kids. www.kidswithfoodallergies.org
They were incredibly helpful to me when our 6yo son was first diagnosed with allergies (when he was 1 yo).
-D.