J.G.
I agree with you that you probably shouldn't eliminate the nap entirely. Your daughter does seem too young for that.
Have you tried pushing her nap back an hour or so and putting her down 5 hours after waking rather than 4? I wonder if the extra hour of activity would make her more tired and more ready for the nap. Perhaps she would fall asleep quicker.
My daughter is 17 months old, and she usually goes to bed at 7pm, wakes up around 8am and naps from about 1-3pm. She's a little younger than your daughter, but she has tolerated being up for 5 hours before her first nap. Maybe your daughter will too. Just try pushing the nap back by 15-30 minutes every few days and see how she does.
If for some reason, she still doesn't go to sleep, I would still use the scheduled nap time as "quiet time" in her crib. I think that even if she's not sleeping, the time spent being quiet and still would be restorative for her.
Good luck, and congratulations on baby number 2!
J.