I think you need to use a reverse proxy such as Nginx for that. Personally, I prefer Traefik over Nginx as it can automatically handle tls certificates for you, so you do not need a seperate service for that. (I've also found Traefik to work better in Kubernetes environments, so you can apply your knowledge in both cases; not relevant if you're only using Docker however)

A reverse proxy allows you to decouple concern for things like path prefixes from your application and the deployment environment.

I recommend creating a routing rule for your prefix and then routing that request via a stripprefix middleware to your service. The middleware removes the path prefix from the HTTP request before it gets to your service, so it can correctly match the request path again.