I had never seen a halti till the other day, when I saw one on a beautiful Belgian Shepherd - a breed known for their marvellous natures. My sister explained that they are very common in Sydney and quite a lot of people use them. Whether it would help with your bully is problematic, as you say. It may be a matter of getting the dog so used to the halti that it gives up.
As for cats peeing, my three-legged cat took up peeing on the bathmat and any towels left on the floor, because she didn't want to share her tray with the new kitten. Result: we now have two trays. You can buy a covered cat tray in a pet shop (they smell a lot better, too) for about $75 or you can go to the Reject shop and get the same thing for $25. With a bonus scoop.
Are you keeping the tray clean enough? If you buy clumping cat litter, you can scoop out the lumps every day or so - into a plastic bag then into the bin. Plus, cats like those crystals; though they don't clump for easy cleaning, they soak up odours like you wouldn't believe!
I also provide a big deep plastic bowl outside for the cats - it's full of compost dirt, which cats like. Stops them shitting in the garden so much.