I am left handed but I successfully use a mouse in the normal right handed manner.
This is a help to me because, being a graphic artist, I then use a Wacom graphics pad and stylus in my left hand.
Now, several models of Wacom pads come with their own mouse which you can set up independently of the conventional one. You can also use the stylus in a similar way to a mouse as well.
If you want to pursue this, I would strongly suggest the Wacom brand as other, cheap makes do not have the same facilities.
This is the sort of thing I mean click here
Not the cheapest but it will do what you want and give you a superb drawing tool as well.