It would seem to me that your internal policy about not divulging applicant names to teachers should be a sufficient argument, even if they don’t agree with it. As school policy, you cannot not abide by that unless the head/board gets involved in changing the policy or establishing some rules for making exceptions. If they don’t like the school policy or your approach to it, then the scholarship doesn’t get awarded.
In similar situations I’ve seen at other schools, the typical approach is to do what you suggest. The committee or others provide a list of candidates to the financial aid office, who then selects the recipient(s) from that group as appropriate and notifies the committee who has been selected. This way, the committee that sets the scholarship's academic criteria are ensured that the recipient has met the criteria and the committee that determines and safeguards financial need decisions protects the confidentiality of the pool, in keeping with existing school policy and best practice standards. It really doesn’t have to be more complicated than that. To select one scholarship recipient shouldn’t require compromising the confidentiality of the entire financial aid applicant pool.
Are there any references or notices given to parents about how the school will treat their fin aid info and outcomes? Whether in the enrollment contract, financial aid letter, or policies listed on the website, any language you include that tells parents how the school keeps their fin aid related data/results confidential would be important to share. It might help to remind/inform the scholarship committee that sharing the list violates what the school promises to parents, as well, which could open the door to liability issues, potentially.
If they insist and their way wins out (which should only happen if the head of school makes the decision to allow the list to be shared), you could take some measures to help insure that what you share doesn’t go beyond the committee. For example, before you share the list of students, require each scholarship committee member to sign a confidentiality agreement that helps protect you and the financial aid program in case one of them divulges confidential information about an aid recipient, making them personally liable for any damages that result from breaches. Even in that case, they shouldn’t share the list with other teachers or department chairs to choose someone. They should keep it confined to ONLY committee members and sign off that that’s what they agree to do. You could even suggest that the scholarship selection meeting must include you and that you would only show the roster/list on screen (not distribute the list to people in any way prior to the meeting) or that you would only provide the list on paper at the start of the meeting and collect/destroy the list when the meeting is over (even this is not 100% risk-free).
There are some resources in the Content tab in School Portal related to confidentiality that might help. When you log into your portal account, click “Content” in the top tab bar and type “confidentiality” in the search to find some material that address privacy and confidentiality issues.
This is certainly an area where it's better to be safe than sorry when deciding whether to share financial aid applicant or recipient lists with others within the school, even when the intentions for its use are good. Very few, if any, requests beyond the admission, financial aid, or business office really warrant the potential downsides of disclosing your aid recipients or applicants, even at a cursory level. Families exercise a great deal of trust when submitting their aid application and the process works best when they believe that the information will only be seen and handled by the absolute fewest people who need to know it.