If the school believes you’ll need a lot of aid, then yes, applying for aid will lower your chances of admission.
I would go ahead and take the SSAT. You might as well see how your kid does. Maybe she’ll have a high score which would indicate she could, if she wanted, do well at a more rigorous school. On the other hand, a low score could indicate she’d struggle in a challenging school. You don’t have to submit the SSAT score after all.
My kid applied TO and is a freshman at Emma Willard. Super happy at an all-girls school. She’s making a mix of grades but the school’s many hands-on learning opportunities suit her well.