OP could you write an email to the real estate saying you would like to get a dog and can they ask the landlord what their stance on pets is? When we applied for the house we are in now it said 'no pets' in the listing. We attached a cover letter saying we have a 5 year old cat who is fully vaxxed, desexed, defleaed etc and would be willing to get the carpets cleaned every 6 months. We got the place. You could say the same or if its an outdoor dog say that you are willing to get the yard professionally maintained every 6/12 months.
How long have you been in your current place?