Wallet client with hoisted account to Ethers 6 provider/signer

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I need to support both JSON-RPC and local accounts in my application. I'm using some other legacy libraries which use Ethers, so I need to convert between a viem WalletClient and ethers Provider/Signer.

This example from the docs works for JSON-RPC accounts, but fails for local accounts.

export function clientToProvider(client: Client<Transport, Chain, Account>) {
  const { chain, transport } = client
  const network = {
    chainId: chain.id,
    name: chain.name,
    ensAddress: chain.contracts?.ensRegistry?.address,
  }
  return new BrowserProvider(transport, network)
}

The problem with local accounts is the resulting provider attempts to call eth_accounts via the public RPC endpoint, which obviously does nothing. I need some way to implement a JSON-RPC provider that uses a local account signer. Is this possible?