Enable using base URLs with path extensions

[Johennes]

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PR description

The SDK currently replaces the entire path of the base URL when forming request URLs. This makes it impossible to use base URLs that already contain path extensions. This change simply appends the computed path to the base path instead of overwriting it which matches what the Go SDK does.

Breaking changes

None.

MSRV

Not applicable.

Testing

Unit tests added.

Async

Not applicable.

[CLAassistant]

All committers have signed the CLA.

[devin-ai-integration]

Devin Review found 2 new potential issues.

[devin-ai-integration]

🔍 Failover region URLs will not inherit the original base path

When a request fails over to a region URL, current is replaced by the URL returned from failover::pick_next (livekit-api/src/services/twirp_client.rs:289-290), which comes from the /settings/regions discovery response. These region URLs are bare host URLs (e.g. https://region1.livekit.cloud) without any path prefix. So url_with_path_suffix(&current, ...) at livekit-api/src/services/twirp_client.rs:234-237 will only append the Twirp service path, without the original base URL's path extension. This is likely correct — region URLs point directly at LiveKit Cloud servers, not through the same reverse proxy — but it means the base-path feature only applies to the primary endpoint, not failover targets. Worth confirming this matches the intended behavior for users behind a path-based reverse proxy.

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[Johennes]

I think this is fine because next comes from the server so it should hopefully already have any required path?