Key for stability is bullet length vs calibre and twist rate.
In 308 Win, 1 in 10” twist is a very popular twist rate. Most 308s will happily shoot 165 and 180 grain bullets Traditional lead cored bullets well enough.
Most copper bullets are the same length as the next size up of lead. So a 130gn Copper bullet will be same sort of length as the 150 gn lead bullets.
Of course thats an approximation as bullets of same weight all vary in length according to brand / design.
The Fox Bullets are a flat based, cone point bullet, so tend to be quite short for length compared to sleek boat tail designs. But as a consequence they tend to shoot well. 130 gn shoot faster and flatter, with less recoil than the 150’s. But the 150’s have bigger thump / energy and will retain energy for longer, but if you look at the ballistics charts there is bugger all real life difference out to 200 odd metres.
With copper bullets no real difference on the smaller deer. They die. With the bigger deer you do need to make sure your bullet goes through the front of the chest cavity where aorta, heart, major bundles of nerves etc all located.
Copper bullets penetrate very well. They don’t fragment / explode like many of the lead core bullets. If you put a copper bullet through the lungs just in front of diaphragm on a large red stag, it may go quite a way before it expires - a lead bullet will make more of a mess.