Client implementation for Cookie auth APIs

[L7R7]

I'm currently experimenting with servant-auth and Cookie based authentication and I would like to write a test for such an API using servant-client.

I have the following API definition:

type Unprotected =
  "login"
    :> ReqBody '[FormUrlEncoded] Login
    :> Verb 'POST 204 '[JSON] (Headers '[Header "Set-Cookie" SetCookie, Header "Set-Cookie" SetCookie] NoContent)

type Logout = "logout" :> Get '[JSON] (Headers '[Header "Set-Cookie" SetCookie, Header "Set-Cookie" SetCookie] String)

type Protected = "name" :> Get '[JSON] String

type AuthAPI = (Servant.Auth.Server.Auth '[Cookie] User :> Protected) :<|> Unprotected :<|> Logout

which is supposed to model a form-based login, logout and a protected resource.
This is all fine and works as intended. If I now try and generate client functions, it doesn't compile:

_ = client (Proxy :: Proxy AuthAPI)

Error:     • No instance for (HasClient
                         ClientM (Auth '[Cookie] User :> Protected))
        arising from a use of ‘client’
    • In the expression: client (Proxy :: Proxy AuthAPI)
      In a pattern binding: _ = client (Proxy :: Proxy AuthAPI)
   |
17 | _ = client (Proxy :: Proxy AuthAPI)
   |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

I found out that servant-auth-client- only supports JWT and basic auth, so it probably isn't what I can use here. I'm also aware that there's ongoing discussion in #1484, but to be honest I'm a bit lost in the details and don't know how to proceed here. Is that something that is supported? Or will it be supported in the future? Or do I have to provide my own implementation for a HasClient instance?

I have a minimal working example for that here. It also includes a test, so you can see the compile error here as well.