I've never had a dog (because of fear of extra parasites and because they eat meat), but I have always thought it would be wonderful to be around such a loving being all the time! They are absolute gifts to be around!!!
The amount of love and joy they bring you must surely be doing wonders for your health and happiness on so many levels. If you can just keep them as parasite free as possible and get rid of the slugs, then surely it would be better in every way to keep them. I really hope you can find a good solution to the slug problem.
Would spraying the whole area with eucalyptus oil or some really hot oil (like cayenne) work? Or could you pave the area and have a border of that copper ribbon around it? Expensive, and not as pretty as grass, but maybe worthwhile???