I have flown many times with firearms and no TSA locks, usually with ammunition in the case. Both are permitted by TSA regulations. TSA sets the rules, but certain airlines like to tweak them, which is why it is best to have a print out of both rules. However, TSA locks on gun cases are not permitted. Despite TSA's claim of a universal policy for firearms treatment, there are certain changes from airline to airline, and airport to airport. If it is against TSA regulations, show them the regs., and challenge any requirement that you change to TSA locks. Those locks are a joke in my opinion. I have had them left unlocked, broken, and lost by TSA agents.
Some other points to ponder. Ship your guns in a quality, heavy duty case, with a good integral locking system. My favorites are Americase, Ziegel, and Bear Track. Pelican also makes a good hard plastic gun case. If you want to travel incognito, put your guns in soft cases, and pack them in a golf bag hard case, using clothing as padding. You will still have to declare your guns. No matter what you use. At one time some of the airlines offered gun cases for rent.
The decision to pack ammunition in your duffel, or gun case is up to you, with the exception of certain airlines. Ask yourself what would happen, if your ammunition was lost. Can you find ammunition for your wildcat .338 Super Whiz Bang in South Texas, Namibia, or the Canadian bush? When traveling, it makes a lot of sense to carry firearms in standard chamberings. A few years back, I went on a handgun hunt in the Texas Hill Country. I brought one gun chambered for a wildcat, but also brought a Savage Striker in .308 Win., and an S & W 629 DL in .44 Magnum. I shipped my ammunition by Fed Ex to arrive before the hunt, but it never arrived. It was finally located at the Fed Ex terminal in San Antonio, and was picked up on the last day of the hunt. Because of the standard chamberings, I was able to scrounge a box each of .308 and .44 Magnum. The wildcat never made it out of the case.
For what it is worth, I have found United, Southwest, Air France, and Delta to be the most gun friendly, with American and BC Air being the least gun friendly. If traveling out of the country, be sure to get proof of ownership from U.S. Customs before you leave.