They commonly use bamboo for what you are proposing in Australia and it seems to work fairly well.
That is very encouraging news.
root mats are fairly shallow (to about 3 feet deep).
That would probably be shallow enough, but I have been led to believe from other sources that most of the mat for Phyllostachys Aureosulcata would be even shallower than that. The main concern is that it should not penetrate the rocky clay with anythig large below the waterline. The dam has a freeboard of about 3 to 4 feet at high water so I should be ok there and that extends for a width of about 15 to 20 feet.
I would suggest planting a lower growing Phyllostachys type of bamboo,
The information I have suggests that the variety I have would typically go 30 feet and occasionally 45 feet and in a warmer climate and optimum conditions maybe more. However I am at the boundary of climate zones 5 & 6 and my bamboo is thoroughly exposed, and the soil is rocky clay. After 3 years the grove that screens the sewage lagoon and virtually never lacks moisture and is almost surely getting nutrient rich water has just barely made 20 feet. 2 years ago all 3 groves were killed to the ground by -5 F temperatures and high winds for a week in mid winter. It came back with a vengence in spring but given the soil and weather here I think my planting may not quite make typical heights for the variety. If it grows tall on the dam, that would be fine with me as it would be a good wind break and it would also provide a privacy screen and dust screen for that part of the gravel road.
I Googled "voles and Missouri" and I came up with woodland vole, meadow vole and prairie vole, so you have voles there as well. ................ They love bamboo rhizomes and I see that they have them in parts of Missouri. ....... I guess that you also have woodchucks there, also called groundhogs.
Well I can attest that we have some sort of mole around here that seems to frequent the garden and the bamboo groves, However in neither case have I found any problem for the plants that I can attribute to them. The ground around here stays so hard much of the year, that I think the burrowing critters may actually be helping the bamboo by loosening up the soil so the roots can spread easier, so it may be a good deal for the critters and the bamboo. For all that; whatever burrowing critters we have on the dam, that does not seem to be an issue. and we definitely do not have woodchucks here. The leak seems to have come from the original gravelly bottom stream, which is providing at least part if not all of the problem. The outbreak of water below the dam is well away from the foot of the dam and in the old stream bed the trees may or may not have caused part of the leak path. And the key suspect is the drought causing extensive soil shrinkage to open up a leak path that has now been enlarged by erosion in the passage way. Taking the trees off will be merely a precaution so that they do not become the next problem, most of them are small. The objective for the bamboo is to keep the trees from coming back and keep me from having to mow the steep dam face while re-establishing the wind, dust, and privacy screen. (wouldn't want to skinny dip in sight of the road traffic, dontcha know