DWP say lodger's rent is income. They take half.

Recently I looked into the effect of a lodger's rent on my benefits. I get housing benefit, support group ESA and both the enhanced components of DLA. My rent is £1750 per month and my Local Housing Allowance is £210 per week, so all my benefits are sucked up by the rent. I asked the council and the DWP what would happen if I had a lodger paying £650 per month. The council said my housing benefit would be unaffected. The DWP said the rent would be treated as income. The first £20 pw would be disregarded and 50% of the remainder would be deducted from either my PIP or ESA (I can't remember which - I'm not sure it matters.)

I asked the DWP if they could treat the lodger's money as a reduction of my rent, not as income, but the person I spoke to on the phone said "no" very firmly. So I'm assuming the matter is not open for discussion and there's no point writing a letter to put my case. It does seem ridiculous that all the money which is supposed to help with my disability goes to my landlord. I'd like to move somewhere cheaper but all the landlords and letting agents in the area don't allow tenants on benefits unless a third party guarantees a whole year's rent. Another option is to move to a cheaper borough but the council said that would shift me to Universal Credit.