I'd step back for a moment and ask several things.

First, what is the projector make and model he currently owns?
Second, where does that projector live in relation to seating?

Knowing what he has allows you to use your google skills to determine fan noise when product is new. Understand that fans can get louder over time, of course.

Understanding where that projector lives currently, plus the spl of the fan, allows you to understand how loud it is where he sits.

You can then find an alternative with lower fan noise, and a longer lens, and give him real world comparison numbers that validate his complaint.