If you’re eligible for Universal Credit as a couple, you can apply for an advance payment to help cover your rent. This is like a loan, so you pay it back with small reductions off future UC payments, but can get you out of a tight spot.
Could you also talk to your landlord and ask for a deferral? They might let you pay a bit late if you can assure them the money is coming in.
I think the benefit you’re thinking of is Attendance Allowance/Carer’s Allowance. This is for someone you look after who already gets benefits (or is over 65). In your case, it sounds like the disability element of Universal Credit and an application to PIP are your best long-term bets for securing an income for your partner.
In the short term, you could also apply to your local authority for some help. They have crisis funds for vulnerable people, so it’s worth giving them a call to ask how they can help.