Yeah, you'll never be able to snap a towel in your throw by trying to snap it like you would in a locker room, although the two are a fairly similar feeling in the wrist when you get it right. If you're not getting it, it's likely due to timing, which you'll probably only be able to fix with experimentation. Start with the hammer pound and try and integrate it into your throw. Try experimenting with some of these things:Note: to me, the pull is from reach back to right pec, even though you aren't actively pulling with your arm.
Pull close and into the chest, don't even worry about the rest of your throw, just pull straight into your self and see what happens.
Use hip and shoulder turn to pull the disc through, let your arm just be a guide into your chest.
Get your weight forward during the pull.
Delay your elbow extension until your bent elbow is so far forward that your wrist starts to coil inward (the "right pec" position.)
Transition from the right pec position to the hammer pound to throw the disc.
If you get the right combination, you will feel resistance on the back of your wrist as you extend your forearm and try and pound the hammer with your wrist. When your wrist finally extends, the towel will snap forward, and you'll know it. It will feel like what you do with a towel in a locker room.
You can also experiment with reaching away from your body instead of straight back, sometimes this makes it easier for you to feel the wrist start to coil inward, especially if you tend to not get your elbow far enough forward after reaching straight back. If you still aren't feeling it, try pulling lower, sometimes you can get better leverage this way.