Should you send refined version of manuscript to editor three months after initial submission?
I would not do it. If the referee is happy with the submitted version, you should add the changes when you send a revised version, explaining what you did with the utmost detail. 

Keep in mind: if the changes are significant, and only to make the proof nicer/easier to read, it might not be worth doing it. Perhaps just add a remark explaining how to do it in a nicer way, or something like that. If you change the manuscript too much, you risk the referee to be very annoyed that he spent a lot of his/her (free) time reading and understanding the paper, and now he has to do it all over again. This is, of course, if the editor is also not very annoyed. 

Minor changes are expected and very common, but major unrequested changes might not the best idea.