The 2014 care act does allow an LA to broker a deal with a care home on behalf of a self funder if they like, the guidance notes to the act say:
Self-funders who ask the local authority to arrange their care
41) The Care Act 2014 enables a person who can afford to pay for their own care and support in full to ask the local authority to arrange their care on their behalf. Where the person requires care in a care home to meet their needs, the local authority may choose to respond to the person’s request by meeting their needs. Where the person requires some other type of care, including other types of accommodation to which the right to a choice applies, the local authority must meet those needs. In such circumstances, whether because the authority chooses to meet needs in a care home, or is required to meet needs in some other type of accommodation, the same rules on choice must apply.
42) In supporting self-funders to arrange care, the local authority may choose to enter into a contract with the preferred provider, or may broker the contract on behalf of the person. Where the local authority is arranging and managing the contract with the provider, it should ensure that there are clear arrangements in place as to how the costs will be met, including any ‘top-up’ element.
43) Ultimately, the local authority should assure itself that robust contractual arrangements are in place in such circumstances that clearly set out where responsibilities for costs lie and ensure that the person understands those arrangements. Self-funders will have to pay for the costs of their care and support including, in cases where they choose a setting that is more expensive than the amount identified in their personal budget, the top-up element of the costs of that setting."
Long and boring but the implications as I read it is the LA do have a choice as it says they "may choose" but I somehow doubt they'll want to get involved.
I don't believe you'd get it any cheaper, why would an home give a place to a self funder cheaper just because you were paying through the LA? If they did you'd just be asked to pay a third party top up to the full rate yourself.
K