Is it permissible to request a new reviewing editor?
As chief editor of a journal, I can say that I would see such a request as reasonable under certain circumstances (and it has happened). However, in the case you describe I do not see what it is that the editor has really done. The reviewers comments form the basis for your revisions and the editor should judge if any comments are more important than others and summarize what the editor sees as important revisions you should make. It is then up to you to do these as you see fit and if you disagree argue against those that need such action. 

Change of editor requests come when an author feel a paper has been unfairly judged (usually rejected) and I know of one case where an editor rejected a paper after using a set of reviewers that were marked as "non-preferred" due to a conflict of interest. I also know of cases where a zealous copy-editor has modified the language to a degree that the science was almost completely lost. In your case, it is not clear what has happened to the manuscript, has it been rejected? have you written a rebuttal and argued against changes imposed? 

I would suggest that you make the revisions you think are reasonable and provide a rebuttal on the suggested changes to your paper. If you encounter more unreasonable resistance from the editor, I think it is only fair to write to the Editor-in-Chief and state that you think you have been unfairly treated and wish to get a second opinion on the handling.