Hire an expensive consultant to tell them what they need to hear. I’m sure if they pay $100k for the privilege of hearing they need HTTP/2 they’ll be more receptive to the need.
I am an expensive consultant. My advice is skip HTTP/2. Either wait for HTTP/3 to ditch TCP or start inlining resources into single HTML documents to reduce round trips.
I'm definitely looking forward to HTTP/3, but I anticipate HTTP/3 having more problems with adoption due to overzealous firewalls and UDP blocking. I expect a longer tail of "I can't use HTTP/3" clients than "I can't use HTTP/2" clients, which means it'll be longer before you can reasonably count on it for performance optimization.