Of course you can use Bullseye in the 303 Brit and any other rifle cartridge. It's a great powder for reduced loads, but I doubt it'll give you the most accurate load and it certainly won't provide anywhere near top velocities. In the 303 Brit, with the 110gr bullet, you could work from just enough to have the bullet consistently exit the barrel, and safely go as high as 15.0 grs. The danger in greatly reduced loads with Bullseye is that the bullet will be loaded to too low a pressure and will not exit the barrel, and then the shooter will carelessly send another bullet down the barrel.
Yes, Bullseye can be double-charged (possibly even triple) as can a few dozen other pistol powders, and that could easily result in dangerous pressure levels. If you are careless, not too bright, or otherwise prone to mistakes, then not only should you not use Bullseye, but you should not reload, and you should probably sell your guns, and stay inside, just to be extra safe.
There are those who say, "only use Trail Boss, it's idiot-proof". It's a good powder for reduced loads, but I prefer to not be an idiot and have more choices.