If their menu boards are copyrighted, it would be the whole design, not just the names of the items and the prices... it's the way the menu board is laid out, their unique fonts, and the photos included. Copying something without permission (to include taking a photo) would be a violation of their copyright, if they have one.
I refuse to do menu board audits through Field Agent because their in-app camera doesn't allow for adjustments or focusing, and with those weird screens on top of the menu board, the photos turn out awful.
I'll do them through Informa/Mystery Researchers because I can use a real camera and make my own adjustments.
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Plan the work. Work the plan.