What is the PH currently? You will need that info to decide whether reducing the acidity is viable and then if it's going to be effective to get rid of buttercups
Horses and grass don't seem to go together well, in my experience. Always end up with docks, buttercups and little else !
Not sure on local application of lime - generally used on the whole field to reduce acidity to encourage grass rather than sorrel etc
I'm sure others will be more helpful, in due course