I would consider the water as a "ponctual" source. Troughout the shower you have the same quantity of contaminated water. So you have always the same quantity of cobalt-60. Of course, part of the particles will stay on the floor and on the skin of the guy, but i have no idea how to quantify this quantity.
Then, according to the quantity of cobalt-60 considering as a ponctual source, you can estimate the gamma dose. For the beta dose, i would use the same quantity than previously, knowing that all the water leaks against the body. In this way it should be easier to calculate the surface beta dose.
Probably you will find the coeeficient of drop area unit to use including beta and gama somewhere on internet (i don't have my book with me).
Last point, because should be difficult to imagine a perfect model, why not considering a 100% efficient wall of contaminated water ? With 5 mm thickness. The quantity of water being constant, you just have to estimate the size of the wall, then use the drop area unit.
I guess it more or less what you have imagined ?
J'aime pas les matins...