It is not possible to load shot into a bottle necked cartridge like the .308, unless you had a specialty case that had straight walls inside. In a normal bottle necked case, pressure would build up before the shot got moving and also there would be no practical way to make a wad fit inside the base of the case. When shooting shot from a rifle, the rifling in the bore will spin the shot charge and make it kinda useless anyway. Patterns are donut shape with a big hole in the middle. And the pellets will deposit lead in your bore. For very close range shots, some straight walled rifle cases like .45-70 could be loaded with shot. Prepare to be disappointed with results. I do have some experience with commonly available Speer shot capsules made of clear blue plastic with plastic base wad that I load into .357 magnum and .44 magnum cases. They shoot poor patterns, only good for a rat in the barn and such.
If was determined to use a centre fire rifle for shooting grouse I'd develop a handload with a single buckshot pellet, lubricated with Lee liquid alox, and sized in one of the cheap Lee push though dies. Press fit in the mouth of the case, and powered by a very few grains of fast pistol powder. At 800-1000 fps or so, That load may serve the purpose without making too much noise or tearing up the bird.