For most apps just use SSE and the built-in code for making HTTP requests (Fetch) instead of hacking up your own client side JS to make requests over a WebSocket. The latency is the same because modern browsers multiplex HTTP requests over a single TCP connection that is left open.
Maybe if you are making many client requests per second there is an advantage to not sending full headers/cookies/etc... on each request but not if you're sending requests in response to user clicks/touches.
Any sufficiently complicated SPA contains an ad hoc, informally-specified, bug-ridden, slow implementation of half of Fetch.