OP, nobody has spoken to you like scum. You have had a great deal of very constructive advice, as well as a realistic assessment of your dog's chances if they are able to find a rescue place. Your 13 year old Staffie will almost certainly be put down if they go to Battersea - they are having to kill much younger, healthy dogs every day.
To be blunt - your 13 year old is not going to be with you for more than a couple of years more, tops. You have two options, you can try to find a place for your dogs which will almost certainly result in them being PTS after spending a stressful and miserable time in kennels, if they go to a pound. If you surrender a dog to a pound, they can be PTS there and then - there is no seven day window. With your dog's ages that seems the most likely outcome. There is a vanishingly tiny chance of them going to a specialist rescue like the Oldies Club - they are currently overwhelmed and have a waiting list. If they can get on to a waiting list, would your LL be happy with that?
Or you can try very, very hard to keep them for the short remaining time of their natural life in your flat, which given that your tenancy doesn't specifically exclude pets seems more feasible.
Again, can you discuss this with the LL? Is it possible to undertake to not have any more pets after this one passes away? Why has your LL suddenly decided that your pets are not welcome? Could you offer a higher deposit/professional cleaning after your tenancy ends/specialist pet insurance for tenants?
Please try talking to Shelter - they may be able to intercede on behalf of you with your LL. Do you have any other Advice centres/drop ins near you who could take up your case? Local Councillor? Are you on the waiting list for social housing?