Ok, I managed it. There are couple of rules to follow:
- Use websocket header 
"WebSocket-Protocol" -> "graphql-ws" 
- After connection is established, send 
GraphQLWSRequest of type "connection_init" 
- After receiving response 
"connection_ack", send GraphQLWSRequest of type "start" with subscription query as payload 
After those steps server is listening and you can send your mutation queries.
Some draft example:
import caliban.client.GraphQLRequest
import caliban.client.ws.GraphQLWSRequest
import io.circe.syntax.EncoderOps
import play.api.libs.json.{JsValue, Json}
import play.api.test.Helpers.{POST, contentAsJson, contentAsString, contentType, route, status, _}
import org.awaitility.Awaitility
 def getWS(subscriptionQuery: String, postQuery: String): JsValue = {
    lazy val port    = Helpers.testServerPort
    val initRequest  = prepareWSRequest("connection_init")
    val startRequest = prepareWSRequest("start", Some(GraphQLRequest(subscriptionQuery, Map())))
    Helpers.running(TestServer(port, app)) {
      val headers = new java.util.HashMap[String, String]()
      headers.put("WebSocket-Protocol", "graphql-ws")
      val queue = new ArrayBlockingQueue[String](1)
      lazy val ws = new WebSocketClient(new URI(s"ws://localhost:$port/ws/graphql"), headers) {
        override def onOpen(handshakedata: ServerHandshake): Unit =
          logger.info("Websocket connection established")
        override def onClose(code: Port, reason: String, remote: Boolean): Unit =
          logger.info(s"Websocket connection closed, reason: $reason")
        override def onError(ex: Exception): Unit =
          logger.error("Error handling websocket connection", ex)
        override def onMessage(message: String): Unit = {
          val ttp = (Json.parse(message) \ "type").as[JsString].value
          if (ttp != "connection_ack" && ttp != "ka") queue.put(message)
        }
      }
      ws.connectBlocking()
      Future(ws.send(initRequest))
        .flatMap(_ => Future(ws.send(startRequest)))
        .flatMap(_ => post(query = postQuery)) // post is my local method, it sends usual FakeRequest
      
      Awaitility.await().until(() => queue.peek() != null)
      Json.parse(queue.take())
    }
  def prepareWSRequest(ttp: String, payload: Option[GraphQLRequest] = None) =
    GraphQLWSRequest(ttp, None, payload).asJson.noSpaces
  }