Op l am so sorry about your current situation, it is heartbreaking and as l have dogs, l know they can be manged well in a house.
BUT l have been a LL for 20 years, l have had lovely tenants who have become friends over the years, and one property currently has 2 small elderly dogs, as the tenants proved themselves to be excellent, for a year before asking to have rescue pets, so l trusted them.
I have also been nearly bankrupted by one tenant, a very nasty man who stopped paying rent, totally trashed my only rental flat, (he rebuilt motorbikes in there on new carpets...oil and all) and when l challenged him knowing he was employed, he threatened to petrol bomb my elderly parents cottage....
I was 23 and had worked 7 days a week, to buy and do up my first flat on my own, since the age of 18, and then worked half the night in addition, to purchase the derelict flat above, and fix it up myself for another year. This was because l was not in a well paid job, and knew l had to secure a pension whilst l was young enough, to work day and night for some years.......
Getting him out brought me to the very edge of bankruptcy, and took me 3 years to pay off the resulting debt to fix the flat...
I had a similar situation 3 years ago, bad tenants left me with 9K ( yes 9K) of damages.... you may think, what has this to do with not allowing my pets, well, would you really add on the risk of people like l have just described, having animals in the property too?
Some pet owners are wholy sensible and responsible as l am, and l am sure you are.......but the LL doesn't know that until it is too late.
the law protects tenants hugely, and offers littleprotection for the landlord, which make LL's much more cautious.
So much needs to change, LL's need to be able to remove problem tenants within a month. I want to see Independent inspectors appointed to assess properties and rent arrears, to prevent unfairness from rogue LL's, and revenge evictions.
UC needs to be paid to the LL, so we are not afraid of the rent( ie the mortgage) not being paid. Sadly UC beneficiaries are twice as likely to default, hence mortgage companies not allowing them as tenants.Paying their rent direct means many more LL will let to people on UC and may encourage more mortgage companies to allow this too.
I would try and find a property where you can approach the LL direct and discuss your pets...not easy l know but l wish you luck.