Before you get too far into it, start pricing out all the stuff you want to do to the Remington and then price out a left hand custom. I think you may be surprised at how close the costs can be. I had the same plan as you until I started looking into it closer and actually put the Remington in my hands.
I wanted to do the following to the Remington to make it fit me and my wants better:
1. Needs a longer barrel to make use of all the powder this round burns.
2. Blue printing action should be done if changing the barrel.
3. New stock. Factory does not fit in my hands at all.
4. New trigger. Has to be light and crisp to avoid pulling.
5. New muzzle brake. Factory one is not conducive to prone shooting.
So after making all these changes I was so close to have a full custom rifle that all I was really missing was the left handed action. In my mind, being that close, it was ridiculous not to go all the way.
Now I'm not saying you have to do what I did, I'm just trying to point out the things that I found while I was at the same point as you are now. No matter what you decide I'm sure you'll have alot of fun with it!