I will offer my experience from New Mexico on American in 2017.
First, all of the meat was boned out and deep-chilled overnight with dry ice (available at MANY grocery stores for about $5/lb). The cost of dry ice can add up quickly as it is heavy - we purchased about 40lbs to get 200lbs of elk to almost frozen I packed everything into two Coleman "Cube" Coolers with wheels built-in - they were purchased from Walmart for $16.00 each and we capable of holding just over 75lbs of meat each.
When I arrived at the ticket counter at 5:00 am, the agent greeted me with a "congratulations on your successful hunt!" She looked at my ticket and my gear/baggage (1 carry-on, rifle case with rifle, a checked bag and the two coolers) and suggested the following: Checking all the bags would cost $35 each for the rifle case and my carry-on. The coolers would cost $50/bag since I was allowed two and they were overweight, which would add approximately $35 per cooler. Since my flight was a two-leg thru Dallas back to DCA, I could upgrade to First Class for $75 on the first leg and get my first two bags for free, the second bags would cost the coach rate of $35.
So instead of paying over $240 for my bags for a Coach ride home, I could pay $145 and ride First Class at least thru Dallas. With a wink, the agent asked me "which bags would you like to be your two free ones?" We agreed that the coolers would be freebie #1 and #2 - since they were a bit over the 75lb limit, I removed a roast from each cooler and handed one each to the ticket agents who were very grateful. My primary agent (female, 40s) said "Thank you so much, this is the first elk I've had since my dad died, he used to take me every year." So, good deeds all round.
Couple things to remember - TSA will almost certainly go through your cooler and re-pack as they will. DO NOT expect to fly with dry ice - it's a no-go. Get everything as close to frozen (or solid frozen) before you go to the airport. Now, moving around with all this stuff was definitely a pain - fortunately we had rented a full-sized pickup, but returning it and getting the bags dealt with in the terminal took a logistical plan. Forget about shipping it, unless a processor can common carrier ship in a frozen state - FedEx estimated over $800 to ship the coolers to my home (largely because I killed the bull on Thursday and it would have to be shipped out on Friday w/Saturday delivery).
And, of course all of this could be out the window given post Covid flying regulations.
Have a great hunt.