If you are going to have something built then go with a rem 700 or clone. The stock options as well as scope base options are numerous and can be had anywhere. I just got my first custom done, a 7mm wildcat I designed. Almost any smith has the tooling to work on a 700. If you were going to do the work I would say go with a Savage and a pre-fit barrel. (Also, not a bad option if you are wanting to save some weight.)
I went with a manners T-4 because I was really trying to squeeze all the accuracy I could out of it but I have shot a couple of really good rifles with HS Precisions on them un bedded even. Easy to take care of your gun that way (clean the trigger, or let dry) and not terribly expensive if you keep it to base models.
As for a barrel, this is where you are going to make or brake the scales. Mine is 1.4x4" then tapered to .875 @ 30". That didn't sound too heavy to me but this is going to be a 15# rifle now for sure!
If you go with a rem I would suggest probably a light palma contour. If you get crazy, build it yourself, and go with a savage, don't go any bigger than their varmint contour and it should not be too heavy.
Hope that helps, (I think that some day I will also have one of these rifles!, Good Luck!)