Reloading .303 British : Well 150 grainers in the No5 is a bust!

That's marginally useful, even on game the size of deer. Something horrible is happening. You could look for loose screws/bands. You could look for some awful dings at the muzzle. You could try seating bullets out farther, as long as possible without hitting the rifling. That could also give more case capacity and room for a grain or two more powder. You might have to single-feed if longer rounds won't feed from the magazine.

You could check your reloading practice too. Don't rely on the powder measure. Weigh each charge with an electronic scale. It's slow but I've found it satisfying. Trim cases to length each time in case they are varying in bearing length on the bullet.

If you can find a bore-scope or gunsmith, try to have the throat examined. Typically this problem will be a throat too long to guide the 150s into the barrel without knocking them out-of-round in the process. This is what seating out further might help. Neck-sizing only can help centre the bullet a bit. The body of the cartridge should align the bullet with the bore. You could also try those exasperatingly expensive copper/non-lead bullets. Copper is a lower density so the bullet is longer and might do what the 174s do. Nosler "partition" bullets also are a bit longer than single-core bullets. Of course, if the 174s work OK perhaps we don't need to worry about it but it does limit the flexibility of a rifle not to be able to have a wide choice of bullets. A crude test could be done by dropping a 150 gr bullet down the centre of the chamber with the muzzle pointed down. You can then use a rod and a caliper to measure how far in the bullet went before touching rifling. If the throat is eroded too far there's no good solution except using a larger diameter bullet or a rechambering operation (rather bad on a carbine...). Bullets up to .312 diameter are fairly common on the market but you may be able to go a bit larger with your bore.

I'd also try cast lead-alloy bullets. You might even try heavier than 174  if that's what the rifle "likes". When you get to 215gr or so, there's not much difference in performance between cast and jacketed in terms of velocity and the cast are much less expensive. They should be much more accurate than the groups you get with 150 jacketed bullets.

One last, desperate, measure: You could try reduced velocity loads. If the problem is damage to the bullet in the throat, launching at lower pressure/velocity should mean less damage and more accuracy. IMR4895 can run well to some very low charges. You should be able to see some effect if you use a starting load of that powder. If you need to go slower still with a faster burning powder, you're probably out of the running for a 150 gr hunting load.