A few more comments, if I may. The link provided by RnR above is a good source of information, however like all databases it needs to be treated with care .....
For the OP's Hyundai I see that the site recommends an engine oil in the transmission. I've never seen a modern-day powershift tranny yet that uses engine oil, it's more usual for a TO-4 or an ATF although other options do exist for transmissions including some types of hydraulic oils.
For example I found that for a particular model of Hitachi that we use the Valvoline website recommended a hydraulic oil that is, and I quote, "fortified with anti-wear zinc additives". Hitachi recommends zinc-free hydraulic oil (check out the spec of their own HN46 oil) and will deny warranty on any hjydraulic component failure if a customer uses oil with zinc anti-wear additives.
Back to the brakes on this Dana Spicer axle in the Hyundai. Obviously they use the axle oil for cooling rather than having a dedicated cooling circuit. As the OP says when the oil heats up after about an hour the noise starts. Could it be operation-related..? I say this because I have seen operators who have a tendency to ride the left (neutralizer) pedal all the time when they are running. With wet disc brakes in an axle the only cooling they are getting is when you completely let off the brake pedal and let the oil get right into the brake pack. Even if there is just a few psi in the brake line from riding the pedal it will be enough to literally squeeze all the oil out from between the plates, and then you have no cooling whatsoever. I saw this one time on a 992G where no-one could get to the bottom of a brake temp/wear/chatter issue when it was discovered that one of the operators (there were 4 IIRC) on the machine was literally huge and even with the seat slid right back found it impossible to get his foot completely off the left pedal. Therefore all the time he was running he was riding the brakes and causing overheating & chatter as soon as the oil heated up. Do you have an infra-red temp gun you can get hold of an shoot the temps of the axles..? I'd suggest take temps at say 10 minute intervals starting from "cold" around the brake packs, on the outside of the final drive planetary carriers, & the differential housing. The last 2 should be far enough away from the brakes to show if the brakes themselves are heating up significantly more than the general oil in the axle.
Reading all this again, it sounds like a rant, but it's not intended to be. My apologies.