Test it! Seriously, your layout sounds quite uncommon and I am willing to bet that your problem is not frequently encountered. My suggestion would be to A/B or user test the two methods and see what works best.
As for a fixed header: Fixed headers are used (especially in mobile view) to allow the users access to navigation quickly without having to scroll back up.
You mention three different columns and all of them have scrollbars. I also imagine that the header sits outside of these three columns. This sounds quite jarring already. Provided your layout actually works (test it) then a fixed header seems to be necessary.
Things to consider:
1) On mobile, if all three sections are scrollable how will the user scroll to the top by touch alone to find the header?
2) Is there access to the navigation through another way besides the header?
3) Is the mobile view big enough to accommodate 3 column layouts? Should those perhaps split into three individual pages?
4) Do you need all the content you have in the header in the mobile view? Are there things you can remove, move around or hide inside an off-panel navigation?
In all honesty, without a visual, your layout sounds confusing and quite hard to use by thought alone. I cannot really have an educated opinion without knowing any more details.
I can't emphasize enough how important it would be to test your layout and of course the fixed/unfixed header with your potential users. You might be surprised with the results.