Mole draining is excellent value for money although may have to be done every ten years or so,but combined with a rudimentary permanent drainage system would give very good results.
I'm sure there must be high wear resistant,low growth rate,resilient grass species , you could enquire from the Grassland Research bods (if it still exists!)
There are also honeycomb plastic mats that allow the grass to grow through and provide an excellent wear resitant surface in wet conditions.
Unfortunately most of these things require dosh in varying quantities.
Lister