Unable to set URL prefix for server API

[majikman111]

First check

• I added a descriptive title to this issue.
• I used the GitHub search to find a similar issue and didn't find it.
• I searched the Prefect documentation for this issue.
• I checked that this issue is related to Prefect and not one of its dependencies.

Bug summary

I am running Prefect in a Kubernetes cluster behind an nginx reverse proxy and when setting the PREFECT_UI_SERVE_BASE setting the UI base path is updated but api base path is not, resulting in a rendered UI with no flow content. Additionally, updating the PREFECT_UI_API_URL setting changes what path the UI requests but not the path the api is actually served under.

Reproduction

1. Install Prefect
1. Apply settings `PREFECT_UI_SERVE_BASE=/root/` and `PREFECT_UI_API_URL=localhost:4200/root/api`
1. Run `prefect server start`
1. Browse to local prefect server at `localhost:4200/root`

The UI renders and api requests are sent to `localhost:4200/root/api` but fail (are being served from `localhost:4200/api`)

Error

No response

Versions

Version:               2.16.0
API version:           0.8.4
Python version:        3.11.8
Git commit:            17f42e9d
Built:                 Thu, Feb 22, 2024 3:45 PM
OS/Arch:               linux/x86_64
Profile:               default
Server type:           server

Additional context

Could potentially be solved by passing a configurable root_path argument to the FastAPI api_app object if available from settings here
https://github.com/PrefectHQ/prefect/blob/main/src/prefect/server/api/server.py#L615

Related to #11472

[mthanded]

Came across this recently. I'm currently rewriting prefix/api to /api using traefik rules. I agree it would be useful to have support for configuring the api to run at a path prefix too.