Websocket API gateway dynamic endpoint URL

In the HTTP api, I can use path variables as documented here - https://docs.aws.amazon.com/apigateway/latest/developerguide/http-api-develop-routes.html

In the REST api, I can change headers, query params, etc... as documented here - https://docs.aws.amazon.com/apigateway/latest/developerguide/how-to-method-settings-execution-console.html

I've seen this question answered previously, but the response didn't work - https://repost.aws/questions/QUNG4xg_hiSNSukzILneVkgQ/api-gateway-websocket-integration-with-rest-endpoint

No such documentation exists for the websocket API and it's strange that it's not supported. I can't map variables required in my API to be in the path or query params with my express.js routing system. It's a major limitation. It seems that the endpoint URL is fixed. Is there any way around this?