In my opinion, you should not have to deal with this kind of thing from anyone. I would tell him that you cannot live with all the lies and break up with him. It is obviously never going to change and you will be fighting for as long as you are with him and always be wondering if he is lying or not. Save yourself the heartache and move on…
I know that it can be difficult to do, but from what I see, it is not going to get better.
Hope this helps,
~Jeremy