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There's no percentage listed, but Bulbapedia says the faster you beat it, the more likely you are to get a "rare item".

https://bulbapedia.bulbagarden.net/wiki/Super_Training#Secret_Super_Training
It's not a percentage, it's if you beat it as fast as possible, the Dawn Stone is the last item in the selection, rewarded for beating it as fast as possible. I don't know what the time frame is
I tried these same things in ORAS. What I found is that not the fastest times award you the best items, but like Amethyst said, the best chances. I remember getting many “best” times and receiving awards like Stardust, then doing a bit worse and getting awarded stones like the Sun and Dawn. Basically, my hypothesis is that there’s no guarantee but faster times have greater chances to earn a stone.

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According to Bulbapedia, (credit to Amethyst for providing the link)

In Secret Super Training, if the player achieves the target time, they will also receive an item as a reward, in addition to a training bag. The more the target time is beaten by, the more likely the player is to receive a rare item.

I could not find any clear percentage in sites such as Serebii and Bulbapedia. I can, however, go off of past experience. In my couple of months replaying ORAS, I have found that acquiring stones in Super Training is a somewhat tedious process. Beating the target time does not guarantee you the best prize (in your case, a Dawn Stone), but as I stated in my comment, the best chances, corroborated by Bulbapedia’s statement.

You should also factor in the other potential prizes for your Secret Super Training Balloon (Gallade). For you, the prizes are as follows:
- Pretty Wing
- Soda Pop
- Stardust
- Hard Stone
- Dawn Stone

Hope I helped!

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I fixed your link -- you just need to leave a line of space between it and text above it. :)
Thanks so much, I was messing with the link trying to figure out what went wrong.