Night Time urinary "control" is NOT the same as regular toilet training. And "pooping" is OFTEN the last thing to be "mastered" during the whole toilet "training" continuum.
So, do NOT rush it.
It does NOT NOT NOT all become mastered, all at the same time.
AND it will not matter, nor will it "make" her "master" toileting better, if she is in a big-girl bed or not.
In other words... staying dry ALL night, or being "able" to wake, and then walk to the toilet in time, takes time. For most kids... this can happen anywhere from 5-7 years old.... without "accidents."
So, try and treat both situations, differently and not with the same expectations.
For me, my daughter still wore a night-time diaper, until she had total night-time control, until about 5 years old. MANY Kindergarten kids are the same way... parents don't just go around advertising it, for fear of shame. BUT this is NORMAL physical/biological development.
I do recommend, using a waterproof bed-pad or crib pad at night.... that way it makes for easier clean up.
Also, for times you are in the car or on the road... I HIGHLY recommend travel "toilets." We have the "On The Go Potty", which you can get from www.amazon.com it is FABULOUS and came in HANDY MANY MANY times, while in the car or at a park and there was NO bathroom in sight, or while stuck in traffic.
Don't worry.... you do not have to expect a child THIS young to be perfect about "night time control" yet.
She is VERY young, to have toilet training AND night-time control to be coordinated in tandem with each other at the same time. And if she's still in a crib, so what.... to me, that is not a problem. Whatever is best for your daughter... every child being different.
Just keep in mind: pee accidents at night WILL happen... it even happens in 10 year olds. My daughter is now 6 years old... but every blue moon, she has an accident. One night, she said was was dreaming she WAS on the toilet... but realize she wasn't when she felt herself get wet. LOL. No big deal. So you see, its fine... no need to punish or pressure the child. Its NORMAL and natural.
Also, some kids sleep SOOOOOO deeply, that they don't even realize they have to pee. So they can't tell you each single time without fail.
All the best, your daughter seems completely normal and doing fine,
Susan