transitional protection on migration to UC

Can anybody help on this? My son was on Income-related ESA and is on the higher levels of PIP. On migration to UC, he is now on a different ESA, and his new ESA plus UC results in a drop of £54 per week. I asked about transitional protection

"By my calculation, he got £489.30 every 2 weeks for ESA, ….. I calculate this as £1060.15 per month, compared with your assessment on UC of £823.41 per month. Is a reduction to 78% of the previous benefits typical? How do I claim for transitional payments for him?"

and the DWP rep said

"We cannot add transitional protection back onto the claim as he is in receipt of LCWRA now on UC."

My son is in the LCWRA category, but everything I have read on the web says that he should be entitled to transitional protection. The only thing that might be an issue is that he has full time live in carers and some commentators suggest that living with somebody (presumably able bodied) might be an issue. However I think it can only be one of language as my son's social worker has confirmed that live in carers do not live with him, they just work there.

If my reading of the advice on the web is correct, can anybody suggest how to get DWP to agree to transitional protection?

Huge thanks

David