For your partner it's not considered recourse to public funds so no concerns on that council house issue.
As others have said have the council provide you with that important letter or report. If your council can be persuaded to give you a property inspection report for free togther with their permission for her to live there then that would be better. If not then just a supporting letter is OK

As a British Citizen you are fully entitled to claim whatever benefits you are entitled to without any detrimental impacts on the immigration status of your partners.
Just be aware that in principle you're not allowed to claim more benefit simply because your partner is living with you and you'll have no problems at all.