RestClient is internally based on jQuery ajax() https://api.jquery.com/jquery.ajax/ and to add headers to your request you can use

beforeSend

parameter:

val resultPromise = client.remoteCall("<http://example.com/post_endpoint>", obj { value = "test" }, <http://HttpMethod.POST|HttpMethod.POST>, beforeSend = { xhr, _ ->
    xhr.setRequestHeader("Authorization", "Bearer token")
    true
})