I have been to Mexico hunting MANY times are here is what I found:
1) It is FINE to pack your ammo in your rifle case. Go to this TSA link and you will see under "Ammunition", point #2 where it states this. It is fine to do the same in Mexico.
https://www.tsa.gov/travel/transporting-firearms-and-ammunition
Print out a copy of these TSA regulations and take with you. Often times you get a rookie TSA agent or airline agent that does not know their own rules and having a copy to show them makes all the difference in those cases.
2) Go to your airline on line and look at their requirements, they may differ from TSA but most of the airlines are the same as TSA. Print a copy and take it with you in case you get someone that does not know what they are doing.
3) Make sure your outfitter in Mexico has all your paperwork in order and has your gun permit for you when you land. If you have all this, it is a breeze. Your outfitter should have had you fill out this paperwork already and submit 3 passport type photo's for your gun permit. One mistake and it will cost you a lot of time getting it fixed. Make sure you declare the number of rounds of ammo you are taking into Mexico on your paperwork and DO NOT TAKE ANY MORE THAN THIS. If you show up with less, that's fine but more, no es bueno.
4) Make sure you get your CBP 4457 filled out and signed by US Customs before you leave for your firearm AND your ammo for your return trip to the USA. The regulations do not state that you have to have your ammo on this form but as of late, Customs has been picky about wanting to see the ammo on the 4457 just like the rifle and scope, etc. even though you cannot purchase ammo in Mexico.
5) I am guessing here that you are flying in to Hermosillo? If you are, here is some advice that your outfitter probably won't tell you. The flights to Hermosillo are almost always overbooked during hunting season in Mexico. The airlines do not guarantee you a seat on your flight until you check in and get a seat assignment. Having a seat assignment alone is not being checked in. This usually cannot be done until 24 hours before your flight. Check in on line as soon as you can to insure you have a seat. They WILL bump you in Phoenix if you have not checked in, the flights WILL be oversold. This also holds true on your return flight. You will be at a camp somewhere and will not be able to check in so have someone back home do it for you 24 hours before your flight. Be sure they have all your information including your passport number, in order to check in for you.
6) Another thing to watch out for if you do not get on your flight. The airlines will commonly book you to another city and bus you back to Hermosillo. Let them know you are meeting your outfitter at the Hermosillo airport and he has your Mexican Gun Permit and this is not acceptable. A group of hunters that I know got flown into another city and was getting bussed back to Hermosillo and when they landed, their gun permit was in Hermosillo with their outfitter. Their outfitter had to drive down and get them and it ate up a day of their hunt.
I have been into Mexico to hunt many times, in many ways it is easier than the US and it is a BLAST to hunt there! You will be wanting to go back!
Shoot me a PM if you have any other questions.