Yes! Walk walk walk walk walk -- Does she have a nice forward flatwalk? Just ride her at that -- the walk will help her time up as it is a 4 beat gait as well. I would walk her (not a slow sloppy dogwalk, but a forward walk) for a while before I would ask for any speed.
As for trotting, as others have said, no it will not hurt her physically. My horse Smudge (RMH) only trots at liberty - a big beautiful trot, but only gaits under saddle. He gaited the first time he was put under saddle, it isn't a taught gait. Most trotty horses make the best rackers. Smudge is a really square moving horse, always racky, never pacy.
I understand your falling in love with your mare's personality. The mountain horse temperament is what sets them apart from other gaited breeds (NOT saying that other gaited breeds are not good, just that the temperament has been bred FOR in the mountain horses for generations).
They are like having a big labrador retriever around, always 'wagging their tail', always wanting to please and be with their people.
I am a KMSHA examiner, if I can answer any particular questions, please don't hesitate to contact me.
BTW, since KMSHA books are wide open, I am curious to know your mare's lineage. If she is not KY Mtn top and bottom, she would have had to be certified before she registered. If she is Ky Mtn reg. By birth, she should indeed have the gait 'in her'. Good luck with her!
Jude in Ohio
start out walking her with a little rein pressure, click her and let out some rein as you go, let up on the rein pressure, if you find a smooth gait hold it, you can let it out some more but when it starts getting rough just apply some pressure to hold the gait. You can do it from a trot gently put a little pressure on her she may fall right into a gait. It takes a feel to get it right but if you should be able to find out quick if she will gait or not. I've ridden some in the past that took an expert to hold a gait (which Im not) and some that just fell right in it and after a while of riding would hold it.