Hi wise gardening gurus 
I'm looking for some advice regarding what to do with one of three lawned areas in our back garden please.....
Prior to buying our 1850s house in late 2014 it had languished on the market some 18 months. Before that the garden had been kept up together by the employment of a gardener and in the property details both the lawns and flower beds/borders were immaculate, but during marketing the vendor bought elsewhere leaving the house empty, let the gardener go and subsequently the plot fell into a state of neglect.
First thing we did on purchase was to add secure fencing to the part walled garden to make the remainder safe for our dogs and last Spring/Summer we began work on restoring the rather overgrown beds/borders, but the two intervening wet winters and our boisterous pair has resulted in the upper lawns becoming somewhat quagmire-like 
In particular the small lawned area between original 1920s paved terrace (dating from when the house was extended) and rose bed is now in need of a major overhaul. Last Autumn we tried grass seed but this didn't take and we're now contemplating returfing it completely. However, my gut feeling is that before long the dogs - who regularly fly out the French windows, down the old stone step, skid across this apology for a lawn and into the main garden areas - will soon ruin our efforts!
Therefore I'm looking for alternative suggestions or ways to minimise the risk. I've added a few pics (may need to put some in a further post) showing the area in question. As you can see from the first pic it is currently pretty bald - significantly worse than it was this time last year. Beyond the Rose bed the garden drops away and at opposite ends there are two further sets of stone steps leading down to the lower lawn. I can't see a way of deterring the dogs from using the small lawned area in question as a shortcut to both lower lawn and other lawned area (seen in second pic). The third pic shows the view from our balcony above, giving an idea of layout.
I've thought about doing away with a lawned area in that position altogether - either by widening the flowerbeds (there's also a narrow one with perennials, shrubs, ferns directly next to the paved terrace) but don't think that would look right......or maybe it would? Or laying gravel/pea shingle instead of turf, but think that would be out of character......
Any suggestions - other than keeping the dogs out of the garden! - would be gratefully accepted.......TIA!