Hey partner! Unfortunately I have never  had the opportunity to hunt/process any of the exotic deer species. But I know of  no reason that it should be handled differently than any other deer, and unless there is a major difference in the feeding habits of an Axis deer, it should not eat differently than other deer taken from the same area.
    The first thing I question is where was his cooler kept and how well insulated was the cooler? Unless his portable cooler was put in a walk-in cooler, I doubt that the ice would hold  for three days at the temperatures in Texas. Even in the north country most hunters that have their food, beer, or venison in coolers check the ice and temperature daily, draining off any water and replacing with new ice. I would guess that after three days there may have been no ice and the water temperature could have climbed above the 40F safety level.
   
    You already know my feelings on soaking meat in water (I don't condone it), other than that I would question the shooting/handling of the Axis deer compared to the whitetails taken from the same area during the same time period.
    It's my guess that there was an area in the hunting /handling process that allowed or caused bacteria growth that led to the smell & breakdown of the meat.
       
Other areas to consider would be:

        1) Where did the shot hit? Shoulder, neck, or could it have been a gut shot.
        2) Did it die immediately, or did it have to be tracked? Excessive adrenaline seems to have an effect on the quality of the meat.
        3) How long before it was field dressed after it died? Any length of time here causes gas buildup and bloating of the cavity.
        4) What was the temperature that day? The higher above 40F the faster the bacteria growth.
        5) Was it rinsed with fresh water after field dressing?
        6) What was the length of time between field dressing and getting it in the freezer?

    Next time why not try putting the whole deer in the controlled environment of a commercial cooler between 32 - 36F. Then after the 3-4 days skin, cut, and freeze it without the soaking process.  If you can't find a cooler to put it in then try cutting and freezing it immediately. I've done it both ways for just short of 40 years with no complaints.