Many landlords find themselves in a situation where they need to accept a financial loss in order to have nightmare tenants move on and avoid longer term financial loss.
Op I think you are in this position now,
You need to think about the reality that you won’t be marketing the property again until these tenants have gone. You therefore want them gone ASAP and will probably need to make it easy for them to go which will probably involve some financial loss to yourself.
The trouble is, although they’ve said they will want to go, it is likely to be all bluff as they know moving is expensive and inconvenient and no one will be keen on a big dog in the small property they can afford. So sadly, I suspect they won’t actually confirm they are planning to use the break clause and go in February or before.....and you will be stuck with them, probably with a dog and probably turning out to be difficult in other ways too.
I assume your break clause applies to you as well as them. The difficulty is with Covid that longer notice periods are required and that might make enforcing the break clause from your end tricky...but I would look into it.
Otherwise, I would give them 48 hours from when you emailed and then ask again how they plan to proceed - will they be calling on the break clause to vacate in Feb or do they plan to stay on the understanding g that the contract applies and there can be no dogs.
I would be friendly in my tone of correspondence. I might say I can see it is disappointing for them if they have decided they would really love a dog and that you can see your small flat is t suitable for a big dog so I’d erstand if they feel they need to move to somewhere that is. I think I might offer them a good reference and some of the upfront rent back if they commit to exit at a specified date before Christmas and confirm the arrangement before the end of October. You’ll need to make it attractive to them....and I think there is still a high liklihood they won’t go. If they stay I think you need to keep reiterating that the property is not suitable for dogs, the tenancy they signed clearly specifies no pets and you will not be able to budge on this. Drop in that the neighbours wouldn’t be happy with a dog either and would certainly be letting you know if one appeared to be in the building.
Yes to asking the neighbour to let you know if there is a dog. It is in their interests too.
Op, did you reference these tenants carefully? It sounds like you don’t use an agency or might be a first time landlord. These kind of problems can happen to any landlord but are so much more likely if you don’t carefully reference potential tenants and know how to deal with a difficulty when it arises to give you the best chance of minimising it.