R.K.
Rather than give up the nap, I would just alter your expectations. Talk to your daughter and explain that her body needs to rest during the afternoon. I agree that running her ragged right before nap probably isn't the best idea. Maybe try that in the morning before lunch. Then after lunch have a quieter routine that leads to "rest time" tell her that she needs to stay in her room quietly...or even her bed. Let her look at books and play with a doll, but she needs to be in her bed/room. If she falls asleep great...but have a set time that rest time is over at. That way you'll stay on schedule. I would try to hang onto the nap/quiet time as long as she'll let you, especially with another little one in the house. That is valuable time! The children I nannied for in college had quiet time every afternoon until they went to school for a full day. Its a great way to establish a routine and give them time to play alone, w/o their siblings input.
This way you'll be instilling a positive way for her to get the rest she needs and even if it is just playing quietly and independently which is an important skill to develop as well. There's nothing wrong with taking a shorter nap or giving it up if your daughter is ready for it, but that doesn't mean she shouldn't rest or play independently during the day.
Good luck and keep us updated on how it goes!