Fixed it. Switched over to a different browser and got a more useful error in the console there:
Access to XMLHttpRequest at 'http://192.168.17.151/api/user/' from origin 'http://192.168.17.151:8888' has been blocked by CORS policy: Response to preflight request doesn't pass access control check: No 'Access-Control-Allow-Origin' header is present on the requested resource.
In the compose.yml file, I appended the port 8888 to BASE_PUBLIC_URL. All is well, now, both when accessing Baserow from my local IP and using a custom domain via a Caddy revers proxy