Isn't clover a legume not grassy hay-type plant?
I wouldn't really dry it to replace hay full-on, but i bet it is an enticing little treat to the buns. I do this to herbs and some other grasses to later sprinkle onto "old boring" hay and it works wonders.
Our patch of land has all sorts of strawwy grassy hay type plants and i harvest&dry them as well, but not enough to get a supply to last through winter. It is also a drag to do this (though dad showed me how to use an electric bush cutter so it is faster, but still...) i get it.
If the hay right near you isn't good enough, have you tried looking further away? Have you tried planting some haygrass yourself?
Besides that, i don't really have any idea myself.