Effect of project coordination task on academic profile
In the comments above, I asked whether OP had any experience performing project coordination tasks similar to that of the position under consideration. OP's reply:


  Yes, I have five years of experience from industry, not directly as a
  project manager, but as a product manager. So it would be within my
  current skillset.


You have obtained one answer here which emphasizes the benefits of gaining this sort of experience. However, it doesn't sound like that answer is applicable to you (though, you would probably know better than us whether this experience would truly be worth it, or if you would just end up doing stuff you already know how to do). 

Let's review where you are, at this critical stage in your career:


You are just now wrapping up your PhD.
You are getting ready to start a transitional postdoc.
You seem to be interested in pursuing an academic career.
You have already gained project management/coordination experience from working in industry.


To my mind, all of the above points to one conclusion: 

You telling your PI/institute director "no, thanks." 

Instead of taking on the coordination tasks, I recommend that you use all of your time in "postdoc mode" working on the things that will strengthen your research profile: producing high-quality research output, finishing up manuscripts, defining your research agenda, growing your network of collaborators, etc. 

When you begin to apply for academic jobs at the next level, or for other postdoc positions after the transitional one ends, you will already be able to play up your ability to coordinate complex projects. So, I don't really see the benefits of playing the project coordinator role right now. The downside of taking that coordinator role on, however, is the main concern: I think it would interfere too much with all of the other important things you should be focusing on at this critical stage of your career. 

(And, let's face it, spending 25% of your time on project coordination tasks for a large project, with lots of moving parts, is probably a very conservative estimate.)