A trademark dispute is a civil issue between two parties. We have legal systems to solve these. Cloudflare should ensure that their customers get timely notification of complaints, and that’s pretty much it.
They store the content of the website on their drive to serve to visitors. Whatever processes lie in the backend of whatever website to fetch up-to-date content from an upstream source is not my concern. They are NOT a neutral ISP, they are providing a service to their customer which includes hosting (doesn't matter if it's temporary hosting because they expire files). From our point of view, it is their IP addresses that are hosting the website. They have all the responsibilities a traditional hoster has, no motter how they try to frame this debate.
The only thing they'll do is forward the complaint to the user. Leaving you with no recourse other than to take legal action before Cloudflare will lift a finger.
Unless there's CSAM, of course.