WOW!! She's a cutie.
I'd definitely go saddleseat with her. She looks to be full saddlebred. If you are really determined, you could go through the saddlebred registry and look for horses of her age, color, markings, etc. A lot of people have done that with rescued saddlebreds and have found matches. Then, you send the DNA into the saddlebred assoc. And hopefully find your horse's papers which would allow you to show in breed classes obviously.
She looks full saddlebred to me. She has a lot of natural talent, and if you did stick to saddleseat with her (which I think you should, judging by her high knee action and high head set), you could get her shod with weighted shoes and she'd be even more of a looker.
Do you ride saddleseat all the time or did you just do it on her at the spur of the moment? Sorry, I am nosy.