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In other words what treat would you suggest as a reward during training, such as when mounting, when your horse hates sugar cubes and even peppermints.
I try cutting up carrots, which she loves, into small pieces but I was wondering if there were any other alternatives.

I've had horses that liked pretzels though not many do. Another treat I've used is graham crackers. Though again, some horses can't get enough and other's wouldn't touch them. I remember as a kid, we had three horses. Two were addicted to graham crackers and the third hated them. When we were handing out treats, the two would get so excited and the third would chase them away from the handouts with pinned ears because he couldn't understand the fuss. I had a wonderful Shetland pony stallion who would do anything you asked for a good old fashioned plain Trisket. The wheat and salt I guess.

If you don't need to worry about sugar, Vanilla Wafers are easy to break into pieces and appreciated by many horses.

I use Hilton's Herbs Herballs...(sugar free) but I don't give treats under saddle (but will in hand). They take a bit of time to crunch but are not large.

I don't like to feed anything when the bit is in, but my old horse loved ginger snap cookies.

Never tried under saddle as I don't give treats with a bit in, but in general, things my horses and donkey like that can be kept in a pocket....
Grapes, but I always squash or bite in half in case they try to swallow whole.
Pretzels, the kind I call "windows" or "rafts" but the bag says something like "snaps". I break in half.
Carrots, sliced up in pocket chip "discs".

Somewhat non-traditional things I've heard other say their horses liked that could go in a pocket... raisins, granola.

Granola bar broken into pieces? I used to make treats for my gelding that were bread pieces with a molasses drizzle, sprinkled with sugar, and baked until crisp.

I've used Teddy Graham crackers and Cracklin Oat Bran cereal, both were quite popular. Current horse loves the little pillow mints and they're tiny enough for frequent rewards.

Check TLkidding's horse shopping thread, I've got my suggestion incl photo in there.

I use a portion of their daily feed allowance. I have hemp pellets (similar to flaxseed) which are treat-size. Or plain whole oats, though they are a bit small and generally a bit more of a pain.

I don't give treats under saddle typically, but Manna Pro has some small training sized treats that would work well for this.

I just use whatever grain I have, but will sometimes buy a bag of the larger size hay stretcher pellets in the winter because they're easier to feed. I generally feed more than one if I'm trying to reinforce behaviors that take some effort, but try to stick to just one if we're working on easier behaviors.

I feed just as many treats under saddle as I do from the ground, as none of mine seem to care if they're wearing a bit, or not.

Why avoid treats with the bit in, hundreds if generations of horsemen recommend treats for helping the horse accept the bit and moisten the mouth. Sugar usually for its clean look on saliva but I cant help asking what the reason is for not giving?

And yes I know all treats should be offered in a way that encourages good behavior, not rudeness.

RP gets treats when I'm mounting and while unlike your horse, he'll eat just about anything I offer, I tend to give him little bits of grain more than anything else. Literally two or three pellets is enough to make the point and most horses will happily take a small bit of their grain eagerly.

What I really like about feeding treats under saddle is that I'm forced to stop and wait while my horse chews, and that makes me slow down and think about what exactly it is I want to reward.

Most of the lessons I've been forced to learn from clicker training have had little to do with the kinds of treats I was feeding, and way more to do with figuring out how horses learn.

The horses where I used to bosrd loved Little Debbie oatmeal cookies. Too much sugar for my current horse.

Mini marshmallows?