I wouldn't do anything you couldn't reverse. The XPR isn't particularly light compared to most modern bolt guns, IMO it is bordering on too heavy. You could try some lead shot placed in a bag then held in place with some adhesive or even Gorilla tape. That stuff holds. This way it can easily be removed or you can experiment with varying amounts to see what works best for you. It doesn't take much 2-4 oz is a lot of weight. And where you add the weight makes a huge difference in the guns balance.
If you look at scope, and scope mount specs there is a huge difference in weight. Mounts can vary between 2 oz to 8 oz. Scopes can vary between 8 oz to 20+ oz. Most hunting scopes range between about 12 oz to16 oz. The biggest advantage of the trend toward lighter modern rifles is that it allows the ability to use bigger, heavier optics without making the overall weight too heavy. You can vary the guns overall weight a lot more by using different optics and mounts than you can by changing, or modifying stocks.