rust-websocket-lite VS tungstenite-rs

You can start by following tungstenite-rs client example. And since you are connecting to an tls enabled endpoint you should enable tls features. And according to your API's doc it requires you to send serialized json string for requesting message response and you can use serde ande serde_json for serialization. This is a complete example where a hardcode string request:

staticpi, is a websocket forwarding service. Basically, it enables one to keep a Raspberry pi, or any computer, “connected”, in order to send and receive messages to and from any client, without having to deal with a static IP address, open ports on your router, or similar. Built in Rust, using axum, which in turn uses tungestine-rs for the websocket connections, tokio, sqlx, redis-rs and others.

There is a client module and client examples. If using tokio, there is tokio-tungstenite on top, which has various client functions.

tungstenite: "Lightweight stream-based WebSocket implementation. It's formerly WS2, the 2nd implementation of WS. WS2 is the chemical formula of tungsten disulfide, the tungstenite mineral."

tungstenite-rs

If you just want to connect to an existing api, async-tungstenite has a connect function. It should work with wss. See some discussion of this issue here.