I hate to say it, there is only one real solution. You only use probably 8 tags 98% of the time, and 5 of those are h1-h5. Ditch the view tab. Its useless. I know it sounds cruel, but replacing that view tab with something that just does a clean preview of what you have in the text tab in an iFrame would be 100% more useful. Then replace the text tab with Ace or Codemirror for code color and completion(lots of plugins doing that already, but compatibility with the view tab tends to be low), put the codemirror/ace input field under a rectangle of that iFrame preview and script the iFrame to be refreshed every time the user presses the enter key in the edit window.
Just like the editing programs we used to use about 20 years ago. I used to be very attached to one named something with "frog" in it. I also had a similar css editor at one time with the same layout that beats everything I have used since.
Has anyone seen anything like this "in the wild"?
As for your question, don't forget to back up first, and depending on how good you are with sql and regular expressions, it might be easiest if you have less than a couple hundred posts to prepare them by hand. Just xerox the sql table and modify them manually is what I would do, up to maybe 300 then I'd use queries and stored procedures, and after that if it is complex i'd just sift for tags and swap them out using php or C#. Regex builds character they say.
Time's fun when you are having flies - the forgotten frog code editor