How recently was she rechecked? If it was more than a few months ago, I would take her back to make sure she doesn’t have a UTI and while you’re there, take the opportunity to talk to the vet about this problem. There are treatments that can help. Cats are like humans as they get older, or like small children, their world will shrink a bit, she may need her bed, litter box and food tray to be in a smallish area (still close to you so she knows what’s going on) so she doesn’t need to travel any distance to any of them.
I’m also wondering whether the surgery on her bladder has simply affected her ability to be continent, maybe the muscles have been affected, or maybe her old age is having the same effect (although 13 is not that old), or maybe arthritis is an issue in her getting in and out of boxes. I would also hazard a guess that she will forever associate her old type of litter box with the dreadful pain she was in then, and that’s why she likes to go to entirely different things that could still be considered a litter box (eg the dog’s bowl). Or perhaps doing it in the dog’s bowl is somehow more comfortable for her, because of her age and/or muscles and/or surgery.
A third possibility is that she can still smell the spots where she urinated when she had the bladder problem and so continues to pee there because she thinks it’s an okay place to pee. Have you used an enzyme cleaner from the vet to clean those spots? Check all over the house with a black light to see where urine has been deposited. After the enzyme cleaning, using the Feliway spray to just spray the spot every day will tell her ‘we don’t pee here’.
In terms of the dog’s bowl, I would be inclined to go with that. Keep the dog’s actual bowl out of the way (of course) except when he’s actually eating out of it but buy a whole lot more (in different colours from the dog’s bowl) and, after you’ve cleaned the spots where she pees with an enzyme cleaner, pop the bowls over the top. It’s not ideal but it’s better than pee all over the floor. Finally, if the vet can’t help you and nothing else works (check the litterbox pink tag, there is heaps more advice), you can get kitty diapers, with washable liners (google for the sites) or you can use (human) pads in the kitty diapers or you can cut down human baby diapers for your cat (check youtube for videos on how to do this). I have a disabled cat who is young but needs to lie down to pee and poop and it took a while to find something that worked for her. Best of luck!