You should get a letter from your landlord to say that he cannot reduce the rent. If you compile evidence that you cannot secure appropriate accommodation for under the limit, you can go back to the rents unit and ask for them to exercise their discretion in your favour - point to all the circumstances that apply, including that you are contracted for the remainder of your lease.
They have the discretion to authorise more than the limits, and a refusal to do so can be appealed, on limited grounds. Landlords however are not legally obliged to wait for their rent while the appeal goes on. You could argue that landlord has implicitly agreed to be bound by the conditions which apply to rent supplement, if they were aware and accepted you as a rent supplement tenant.
If your landlord is prepared to reduce the rent slightly, even if he can't reduce it to the amount that they are looking for, then you'd be in a much better position. Similarly, if you request that you be allowed to stay in the property, only until the tenancy runs out, you are giving the rents unit more and more reasons why they could, or should, exericise their discretion in your favour.