Common powders used in bottle necked rifle cartridges are generally much slower than those used in straight wall pistol cartridges.

Aside from powder burn there is also geometry. The case volume of that .308 is relatively large compared to the bore volume of a 5 inch 308 barrel. Handgun bores trend fat to take more advantage of the slower acceleration times in short barrels. Rifles trend skinny to get more sectional density, and tend to have greater case capacity to keep the pressure curve higher for longer.

With all that said, 308 is going to be severely handicapped by a 5 inch barrel, but it is still running a 55kpsi max pressure with a great big case. It should be at least slightly faster than a 7.62 BLK out of the same barrel length. It will almost definitely deliver better than handgun performance, provided that you can find or make a .308 caliber projectile that provides good terminal ballistics in the velocity range you actually get.

It will however be extremely inefficient. Most of the excess energy offered by that .308 cartridge will go into massive sound and flash. You'll be wasting much of the cartridge's potential for only minimal performance gain over something better suited to the short barrel.