No pets alowed in the Garden!!! Alone she wouldnt have the ability to pass the SeeD test, esspecially with the no gf training rule in Galbadia Garden which is the closest garden to her.

Even if Ulti did say SeeD 6 times as a way to reference to there being 6 SeeDs present (which i highly doubt) it's not like she's all knowing. Maybe she just assumed she was SeeD.