I never understood this current fanaticism for bang-flop over avoiding meat damage. Back when I was young anyone who hung something in a butcher shop that looked like it was hit with an RPG was subject to local humiliation. I guess bang-flop becomes more important when ranges run out to 500+ yards. It might be tomorrow before the shooter can get to the downed animal.
In over half a century of eating more than a hundred deer, elk, and moose that I've shot with 30-06 cup-and-core lead bullets I have yet to encounter a piece of lead in the meat. Absolutely no concern if I did. The potential lead poisoning from consuming the odd tiny shrapnel would not exceed natural everyday environmental contamination. Perhaps it might be more concerning if I was an infant instead of 200 lb adult.
Interesting that I just concluded debating this subject with my African PH. Shots at plains game can involve some distance. For over fifty years I shot North American big game with 180 gr cup-and-core lead bullets and only lost one animal (lightly hit that someone else subsequently killed). Only one wounded animal required any amount of tracking, also lightly wounded. But that one was out of range and shooting at it was a youthful mistake. Anyway, due to component shortages for upcoming trip to Africa I am stuck working up a load with 30-06 165 gr Hornady cup-and-core. I have about a dozen 165 gr Partitions left that I loaded for my first trip and seven 168 gr Barnes TSX bullets from second trip. No hope of getting any more Partition bullets and, though more Barnes bullets can be had, I was concerned because they didn't group well for me last time (I used up most of the box at the range). But that may have been due to operator error (failed to seat the bullets deep enough for a jump to lands) and/or equipment failure. The Partitions performed fabulously on the first trip but I was shocked at the meat damage to the only animal I saw skinned (mind you that gemsbuck was shot at extremely close range). Should I stick with Hornady, try to make the remaining Partitions last, or drive four hours across the border to pick up more Barnes bullets? My PH said don't worry about meat damage. They want clients to put the animal down, hence instructions to shoot even little impala and blesbuck square in the shoulder. My preference has always been to hit animals behind the shoulder to avoid wasting meat. He said the Hornady bullets will work but be careful with Barbary sheep. Interesting that exit wound is NOT desirable for them. If the sheep are spooked their inclination is to pack together tightly. For his last sheep he declined to use his 270 WSM and instead borrowed the landowner's 22-250 to pick off the ram from the flock with reduced risk of bullet exiting. If the Partitions will group with the Hornady bullets, I may take what's left of them along just for sheep. According to my PH the Barnes bullets aren't desirable for sheep because of their tendency for pass through.
Moral of the story is bullet choice is about balancing species, type of hunting, range, and meat damage. I was raised to put meat damage at the top of the list. Bang-flop was never even a consideration in those days. Any animal shot in the boiler room with 165 or 180 gr won't go far before falling over. I have never seen one go further than fifty yards. The OP is hunting small deer and consequently should be concerned about wasting any of it unnecessarily. If he is hunting over bait, which is very common in Texas, he has no business shooting RPG exploding bullets. Long distance shouldn't be a factor. Shoot that zipper with a heavier more solid bullet at slower velocity. Get a good thump without the exploding meat. If the OP is beanfield shooting long distance, then perhaps copper is better. He wants range and more potential for pass through with better terminal velocity. Neither is my typical hunting style. I prefer medium to short range and often as not hunt in snow where tracking is easy. I shoot a medium to heavy 30-06 bullet at medium to high velocity. I rarely have to shoot an animal twice and meat damage is typically quite minimal. But I hunt a lot and I'm a fairly good shot. I have enough experience to know how to be fussy with shot placement.
I also started out as a meat hunter. For the last 50+ years my main sourse of meat was from the deer, elk, etc that I shot, so learned to prefer the "just behind the shoulder shot." I shot my second best 6x6 bull elk in the A-B Wilderness just north of Yellowstone NP. I was actually in there hunting a Bighorn ram as that area was one of Montana's Unlimited ram units. It was my first Unlimited ram hunt, I had never been in that unit before, and I had backpacked in there by myself.
It was the third morning of a 3 day hunt and I hadn't seen any sheep. Shortly after I left my tent I heard an elk bugeling and coming toward me. I was hunting with my .257 Ackley with 117 grain Sierra GameKing bullets. Elk season was also open in the Wilderness portion of that unit then, so I just sat down and waited for that bull comming to me. I first saw his dark, ivory tipped antlers and I knew then that my sheep hunt was over.
It was fairly open lodgepole timber and when he stopped broadside to bugle about 75 yards in front of me, I put one of those 117 grain Sierras just behind his front shoulder. He instantly dropped dead. When I dressed him I found that my bullet had hit a rib going in, pretty well shredded his lungs, and didn't go through the ribcage on the other side.
It wasn't until my trips to Africa when my PHs told me to shoot on the shoulder that I started doing that. When I shot my Leopard in Mozambique with a 168 grain Barnes TSX bullet from my .300 Weatherby, I was more concerned with breaking his shoulder than I was in saving some meat. He died 18 paces from where he was when I shot him.
When I shot my Aoudad in Texas, my guide kept telling my that they were hard to kill so to shoot him in the hams to break his legs. I was again useing my .300 Wby and my 168 grain TSX bullet through his shoulders only allowed him to run about 5 paces.
I started using Barnes bullets in 2010 and since then I have used them in alll of my hunts with my .375 RUM and my .300 Wby except one. Those hunts totaled at least 30 animals, and I was only able to recover 8 of the Barnes bullets. They were all the classic Barnes mushrooms. I think most of the others went completely through the animals. Of all of those animals, only two required a long tracking job, and I blame poor hits rather than bullet failure on them.
The two bull elk that I shot with Barnes bullets fell at the bullet impact and never got up. I shot the first bull at a little over 100 yards. He was standing broadside and I but the 168 gain TSX bullet just behind his shoulder. It came out just behind his other shoulder and he dropped instantly. I was very suprised at how little meat loss there was, especially compared to the meat loss from all of the elk that I had killed previously with 180 grain Partitions.