Well as you know, consistency of a "routine" is important.
But I have also found, that when a child is about to meet another age-set, they have trouble with their usual sleep habits.
Just stay constant. It will pass. Both my kids went through that too... but my son, who just made 3, still naps, and my daughter was still napping up until the first half of 1st grade. They were honestly tired. I had a routine, every single day, and they knew it.
If you daughter will not nap in her room... after her wind-down pre-nap routine... then perhaps, just let her nap on the floor. For a brief moment, when my daughter was a toddler, she went through a phase where she didn't want to nap in her room or in a bed. BUT... she WOULD nap, if on the floor of the living room or on the sofa. So I let her. That way she actually slept. Then that "phase" passed, and she went back to napping in her room. So in other words, even though I had a ROUTINE, I "allowed" her to nap on the floor for that moment of her phase.
And yes, she may very well just be "over-tired." Over-tired kids or babies, actually have a HARDER time, falling asleep, and actually sometimes get more hyper.
... thus, try adjusting her 'nap time.' Make it earlier or later... and see how that goes. For me, if I miss the window of opportunity for naps (ie: when my kids ARE tired), then they have a harder time to fall asleep. But if for some reason I delay my son's nap time, then he gets a 2nd wind so to speak, and actually "wakes up" more.
But yes, I have a REGULAR nap "time" for my son. He is napping now in fact.
Perhaps, she is just "too" active in the morning? Over stimulated children/babies, get too much stimulation, and this just works against them.... and then they can't sleep efficiently or as needed.
There is a myth sometimes, that keeping them real active will tire them out... but many times, this just does not work. When my kids are OVERLY active or had a super stimulating morning... this is when they "fight" naps, more.
Next, I would NOT wake her up early... on purpose, with the hope that she will then tire out and nap, later. Nor would I let her sleep later at night... this just works against them being "able" to sleep well. Just put her to nap and sleep at regular times. And let her wake at her NORMAL time.
Next, if just having a "quiet time"... make everything 'boring' and not turn on anything, nor engage in activity. MAKE IT quiet and boring. For me, I even told my daughter, that "Mommy is just going to sit down and read...no talking..." and that was enough for my daughter to quiet herself and not engage me in 'activity" just to delay things. For me, that worked. Or, I would tell my daughter that "Its Mommy's nap time too.... and brother is already napping... so..." then that was it. Otherwise, my daughter is prone to just talking and talking and not quieting down.
Turn OFF anything stimulating... prior and during nap "time." The less stimuli the better, or the get distracted.
For me, I never used bribery/rewards for naps or sleep. It just is.. and this is/was my kids routine, since they were babies.
All the best,
Susan