I've hunted Pheasants for half a century. Over that time I have had my own game farm license and raised Pheasants along with the helping of my local Sportsman's club raising Pheasants. I've yet to ever hear a hen cackle. But many times when flushing birds, hens come up with a cackling rooster, or the rooster, even tho it cackled, did not flush with the hens, instead it ran the corn rows and flushed farther down. So while cackling will tell you there is a rooster present, one still would need to identify the bird before pulling the trigger.
The OP does not tell where he is hunting, so knowing if there are other species available to hunt or if there is the option of shooting hens(Shooting preserves ans some public release areas around here allow it). My problem with the inability to identify a bird in flight, would also be the inability to identify something else downrange. While me loves to hunt, I would tend to give it up as opposed to just shooting "blind". While I love to hunt Pheasants, and nuttin' beats the sight of a long-tailed rooster flushing outta fresh fallen snow in front of a staunch point.....I also love to woodcock and grouse hunt. Both species where identifying the sec is not necessary.
My suggestion to the op would be to do what he is doing now. Go Pheasant hunting with a friend that can call out the sex, find another species to hunt or stay home. Sometimes, tho, just working one's dog, even without the option of shooting, is a sport into itself.