Supported accommodation is not subjected to Local Housing Allowance - the usual upper amount of HB or UC housing costs payable for private renters - because it includes elements of care and supervision. The rule is not that tenants get twice the usual LHA - it just doesn't apply. In your BF's case the rent just happens to be 2 x LHA.
LHA is incredibly low. It's capped at the 30th percentile of rents for the local area. More accurately, it's capped at what those rents were in 2016 because it's been frozen since then. In high rent areas further caps apply.
LHA will not cover the costs of private sector supported accommodation.
YANBU OP, the whole housing situation in the UK is fucked, along with the HB and UC housing costs system.
They don't want to provide any help to people with mortgages because they don't want to be seen to be helping you acquire a valuable asset. You have to filter it through a buy to let mortgage and a poor tenant before that becomes respectable. Most rage-inducing is when you see this happening in a former council property.
The feeding frenzy over 'unregulated supported housing' for both vulnerable adults and for children is horrific.
www.insidehousing.co.uk/news/news/housing-associations-keeping-thousands-in-unregulated-supported-housing-says-report-64088
www.bbc.co.uk/news/uk-49673455
The answer is to build a fuckton more proper social housing and bring care services and supported accommodation back into LA control. Poor people on benefits, vulnerable children and adults - clearly these people are not profitable in themselves. Anybody making a profit from them is sucking it from the public purse.
And yes they should pay mortgage interest. It's hardly going to be cheaper for them if you are made homeless and they have to put you in temporary accommodation (another category of private rented housing not subjected to LHA - kerching!) - before even considering the effect on your mental health and the associated additional support costs.