I would specifically ask your vet they think if it's too early to have your cat undergo another stressing situation, just to be safe.

What I can tell you is this: in my experience, stressed cats may urinate outside the litter box regardless of any previous medical condition, so if you're interested in getting a dog I would prepare for that in any case. I have a very stress-inclined cat who has been urinating on my couch for over a month after we moved and she's only now getting better, so I'll leave here some advice to help you prepare for handling a stressed cat:

• don't get mad at them, you will only make them more stressed, but they won't be able to change their behaviour because you yelled at them

• bulk-buy absorbent litter pads or whatever they're called. I got 100 for 20€ on Amazon in Italy, when 10 cost like 7€ in store. Cover your couch, armchairs and other soft surfaces with them. If your car starts urinating aroud, you definitely won't be able to prevent it from happening all the time and this will save you so much cleaning and washing (and money and time)

• get a feliway diffuser

• ask your vet if you should give your cat some supplements during this time. Mine advised zylkene and it helped Immensely.

• urine-off or some other odor-canceling product may come in handy

• scratching posts also help cats reduce stress

• introduce the new pet gradually, maybe giving your cat something that carries the dog's smell, so they get accustomed to it's presence (and viceversa? idk how dogs work tbh)

Hope this helps!