I am a massive BARNES TTSX fan . Why ? Because they have been at it the longest and have made the mistakes years ago and many brands are still making.
I have killed all UK deer ( except fallow with these ) in good numbers and prefer the 100 grain for all uk deer out to good ranges even with big stags the bullet will punch a hole in both sides and it expands 100% . This is through my .260rem. I have also shot many with the 120 grain version , in my experience it can become none expanding due to the greater mass and a 200 fps or so speed.
I have also have done a fair bit with the Hornady but only in the 50 grain it works ok as does the 50 grain TTSX when i use my smaller 223 rem . I have also done a shed load of rabbits with the Barnes , its very good on the head shots ( chest shots means a wasted carcass often they are too messy to carry back ) That said Raptors etc get a feed without any risk of lead poisoning ( when by rights all carcasses should be well buried or carried off to a suitable disposal. Likewise Foxes
The only downside of copper is the cost