I've actually heard a little differently- that rice bran is highest in phosphorous, and wheat bran is slightly lower. I actually feed both (but not in the same day). But sometimes I will give my horses about a cup of rice bran, and other days I will make them a wet mash with some wheat bran (doesn't even need to be hot, cold water works well too).
Wheat bran makes good mashes, and rice bran is more of a fat supplement with very little fiber to it. I don't think the term "stabilized" has anything to do with the calcium to phosphorous ratio (I don't think anyway). I think stabilized means they did something to it so it doesn't go rancid, because apparently with the high fat content of rice bran it can go rancid.
I actually started feeding brans to ADD phosphorous to my horse's diet, because I feed straight alfalfa hay. I was aware there was an imbalance with the alfalfa, but I really didn't take it seriously until I found out my mare was in foal, and when I asked my vet about her diet, he said to give her a couple pounds of grain a day to give her some phosphorous since I was feeding straight alfalfa and it is too high in calcium. So I guess I have the opposite problem than people who have pasture or grass hay as the main diet.
So I do give her a few pounds of grain, but also a small amount of bran a day, like a cup or so. And my other horses hardly get any grain at all, so I give them a little bran (of either variety) each day too.
Does this sound sort of okay? I hope so. Broodmare nutrition (and horse nutrition in general) can really make me neurotic about what I feed. I think the feed store people roll their eyes when I come walking in, because they know I will take about 1/2 hour to figure out what I am going to buy there!