The fact that she doesn't reject him right there, she either returns to the Main house, or leave the Annex for disobeying him. If she refuses, she has nowhere to go. Honestly she should leave, but it's understandable that she would see both as bad options.
Leaving is actually the better option (and honestly a bit of a stretch to call "bad") because in her own thought bubble she was getting ready for it after graduating. Implying that she had a worst case scenario ready due to the fact she isn't able to summon a spirit, which is reasonable.
Not to mention her family now can not afford to lose her now that she summoned the rarest spirit. If it were me, I'd even leverage this fact, tell them that if they try anything to me or my friends I'm coming back to burn this entire house down like how'd he burned my hands.
Like what is he going to do? Kidnapping? Well hope the people they hire don't wound or "devalue" the target?
Kill her? Whoops you lost a once in a life time person in history, I'm sure the kingdom/religious organization would be VERY pleased at that, as if they aren't already talked down by the nobles already for the scandal of disowning her.
Lock her up? You really want to have a magical fisticuffs with a legendary phoenix? Really?