you are in a center, right?
Your center doesn't have any accomidations for children who no longer nap?
I would think there may be a few in a center.
In WI, children who don't fall asleep after 1/2 hour (presumably those 4 year olds), are to be offered quiet activities. I would suggest a quiet room with lights low for non-nappers. If she is the only one, have her lay down "until the big hand is on the x and the little hand is one the x", and then take her in the other room and give her some quiet activities, her "work". If she talks to you...say "it's quiet time, I have my work to do, and you have yours,. It's not talking time". Build a sense of camraderie vs. antagonsim.
sleeping is like "you can lead a horse to water..." You can make them stay in bed/on their mat, but you cant MAKE them sleep.
As for the post-nap grumpiness because she didn't get a nap, I am fairly certain that all the fighting about napping in the first place probably adds to it. If you can get something worked out, you may find that she is a lot more pleasant in the pm.