I would say this would only feel justified if the product pricing page already had clear tiers outlined for paid support. Putting a price per hour on customer support otherwise would make me feel like I am just being milked behind closer doors for more money, and non-paying customers are getting shafted. If a paid customer support tier is something you offer, imo it should be clearly outlined on publicly available pages with explicit explanation about the differences between free vs paid customer support. You suggesting it in private communications only would feel suspicious and shady to me as a customer.

However, if the customer themselves suggested to pay you extra for that personal support, that’s a different story.