I wouldn't push her. Leave her till she starts escaping or gets too big for a cot. Why not? Shes obviously happy there and as there is no danger to her hurting herself by scaling the sides I don't see why she should be moved if she is happy.

What we've done with all our children is at the age that they can start climbing  properly and were stable enough on their feet which around the 11 month old mark  we allowed them to get on our bed. we encouraged them to climb on our bed, and played with them on the bed, so that they would feel comfortable on the bed, and sometimes they fell asleep on the bed ( when it was nap time ) and they woke up refreshed and happier and they wanted to sleep in our bed all the time, as they didn't have the bars on the cot to hold them there, so the older 3 were in beds when they were 18 months, and then the eldest girl was in a bed at 21 months, then youngest girl was about 22 months old. They all loved their beds, and still do.

Ava probably could've gone into a toddler bed long before we did, but we thought if she was happy being in the cot, why rush things (she never used to climb out either). She went into her toddler bed at quite an older age, about 30mths. We just decided to take the rails off her bed one day and she was fine, no problems at all (has only fallen out a couple of times). If you have a bed ready, maybe give her a go in that. If she doesn't like it, you can always put her back into the cot and try again later.

We transitioned our daughter as soon as we started to potty train her (she was at 26 months). My daughter was like yours, showed no interesting of climbing out of it.  I think I would have waited a little longer if I didn't start the potty training then.

We have a portable cot for our 2 year old, and because we want her to go into a single bed very soon, we have cut one of the sides out of the cot.  This is so she can get used to getting in and out of the cot on her own (porta cots are much lower to the ground than regular cots) and hopefully the transition to a single bed will be a lot smoother.  You could try a trundle bed for day time naps to start with and when she is used to that, keep her in it over night.  Once she is happy with that, you could then try the regular single bed.