Most benchrest competitions have a "sporting rifle" class. The rifles are usually weight limited, between 9lbs to 15lbs depending on the competition, sometimes a light barrel is required, sometimes heavy barrels are allowed.

And besides that point, there ARE factory production rifles that can be accurized to compete well even in 1000yrd competitions (not win, but at least compete-especially in light rifle competition, 17lbs or less). Remington 700P or the Savage 12 Benchrest being 2 of them. Get it in 300WSM, or 6mm-284. Both cartridges are competitive on the range, as well as effective hunting cartridges.

I personally use my own Savage 12 in .300WSM both for deer and coyote hunting AND for 1000yrd light rifle competition, as well as NRA benchrest competitions. It's 16.6lbs unloaded in the benchrest stock, but it shoots a 8" group at 1000yrds. I have a sporterized stock for it that I use while hunting (drop in action) that reduces the weight to 12.8lbs, off hand shooting is a bear but it's still possible. The only downfall for hunting is that my 12 is a single shot, but practicing reloads is a necessity for hunting AND competition, so it's not a problem.

If you want it bad enough, it's possible.