‘Trapped in an unsuitable property for the foreseeable future’

I would like to express my concern over the huge lack of larger social housing within the area. We are currently in a 3 bed, with 4 children, 3 of which have disabilities. We have been entitled to a 4 bed since September 2021, and are in gold band. Our position on the list never seems to change, and we have only been able to bid on one suitable property, which was in 2021. No suitable properties in Frome have become available since, it seems never ending. The option to go private is also unavailable as the local housing allowance is only £996pcm, and the lowest priced 4 bed on rightmove today is £1495pcm, a shortfall of £500pcm. For a family in a low income this is completely impossible to finance, leaving us trapped in an unsuitable property for the foreseeable future.