What are some of the challenges faced by organizations that use an hybrid model of both predictive and adaptive PM methodologies?

But your questions was about "organizations" not about your organization or a project in particular. If you like an answer in that sense then you could put more detail and we can post our answers related to that. In general, there is no more challenge than when you try to introduce something inside the organization so they will transform the organziation as a whole. So, before to introduce something, and architecture analysis must be done to evaluate the impacts then think about cost-benefit decision where cost is not monetary only.

It sounds like you have one team running iterations around a circular track and monitoring their laps while the other team is running a marathon where no one gets credit unless they finish.

I'm curious if there is someone higher up in your organization who monitors the work cycles within the organization. There are those who would argue that agile frameworks (like Scrum) are ineffective unless they exist and operate within an agile organization. A predictive life cycle project ("waterfall") can exist within an organization, but I would have to ask why they purposely follow a predictive model (plan and run a marathon) when they know they're paired with a team running an iterative cycle (laps around the track).