Based on what you have said so far I think you are ok to include the partner and continue with the claim. If he has just arrived at the UK, applied for leave but currently does not have it I would contact the Home Office and find out exactly what the partners current status in the UK is. If the customer has no status in the UK currently he will be exempt from the NINO requirement:
From Immigration and Asylum Act 1999
115 (9)“A person subject to immigration control” means a person who is not a national of an EEA State and who—
(a)requires leave to enter or remain in the United Kingdom but does not have it;
HB Reg 4(c) Section 1(1A) of the Administration Act (requirement to state national insurance number) shall not apply— to a person who— SI/2009/471
(i) is a person in respect of whom a claim for housing benefit is made;
(ii) is subject to immigration control within the meaning of section 115(9)(a) of the Immigration and Asylum Act;
(iii) is a person from abroad for the purposes of these Regulations as defined in regulation 10(2); and
(iv) has not previously been allocated a national insurance number.