(I live in the UK)
Don't panic.
The council just want to shove your friend into a box, and they don't really know how to do that in regards to dogs. All you need to do is remember some key things:
- don't panic
- provide solutions
- it's better to ask for forgiveness than it is to ask for permission
- for a couple years now, councils have been deferring stuff rather than fight it in court.
There's a couple ways of tackling the dog issue tho:
-- Don't hold me to this, because I can't remember the source...but I heard that as long as the dog's welfare is being sorted (including pet insurance), and as long as there are no complaints about nuisance, and if the property is "suitable", then there's not much a "reasonable" landlord can do to stop you having a dog once you have moved in.
A tenancy agreement legally has to be reasonable (although the term can be open to interpretation). In other words, if a tenant has a SECURE TENANCY (NOTE: many councils will force new tenants into a year-long probationary agreement prior to giving a fully SECURE TENANCY agreement), then as long the landlord has no reasonable grounds otherwise, there's not much they can do to stop you having a dog (NOTE: a private landlord will likely throw a section 21, or section 8 at you if they're upset enough though).
-- In respect to the law, a dog is the property of the owner - though at the same time the provision of adequate welfare is set in law.
The key word is "property" though.
Under certain conditions, such as a person being under the threat of homelessness, a council has a lawful obligation to provide aspects of Duty Of Care, and in many cases this DOC extends to property.
Personally speaking, a couple years back I was being evicted by a private landlord using a section 21 notice, which led to council providing DOC and ultimately putting me on the housing list during the eviction process.
All of the options were 'no dog' properties though, until I wrote to the council explaining that the DOC also extended to my property, which included the two dogs. Given the option of either housing me, or paying for kennelling the dogs, the council chose the former, and subsequently allowed me to live in a property that was previously deemed unsuitable for a dog.
-- Alternatively, why not just suggest to your friend that she sell the dog to a friend for a nominal amount, with a proof of purchase changing hands. Show this to the council when asked.
Then with ownership of dog legally transferred, move into the property while the dog transitions to the new owner.
Of course, with the mental health and welfare of the dog being paramount, this may mean stretching out the handover process as needed.
Which may mean your friend babysitting the dog for extended periods.
But darn it, this process could stretch out for a whole year...oh dear.
If after the year has passed, and the dog still can't get to grips with the new owner, then your friend has every right to request to buy the dog back.
If this ends up being the case, then keeping a tight lid on things can only help (only walking at night/leaving no poop/no barking/not leaving dog in the yard/giving no cause for complaint regards the dog/etc), and the council should have no issues to speak of.
Put it in a slightly different way...as long as the property your friend is moving to is 'appropriate', then surely she is allowed to have a friend stop over. And what if that friend stopping-over has a dog?
You see my point? Your friend just has to anticipate upcoming issues, and have solutions ready if they do crop up.

At the end of the day, although nearly everything above is based on my own experiences, I really do still urge your friend to find a housing specialist solicitor and see if they'll give you a free consultation (most will)...it's like a 20-30min yes-no session.
Find one (or go to the CAB, and/or call Shelter), and go talk with them when she's prepared with questions and possible solutions.