Most tend to go with a DBM due to the overall 3.68" COL of the 338 Norma Mag (same as 338 Lapua). The Norma does have a shorter case, so if you can seat the bullet a little deeper, I would think the std. Rem 700 LA bottom metal would work, provided you get the box, spring & follower used on the 338 Lapua police model. That set-up should allow you to get two in the magazine.
makes a DBM that should work well for you. The magazine release is not a lever that sticks out, but rather a button on the side by the trigger guard - kind of like the AR-15/M4. He also sells a three round magazine, which hangs down just a little.
Another alternative: just have the smith glue the follower to the bottom of the receiver and make it a single shot.