studio: add subdir to live metrics post messages to support live experiments in monorepos

[mattseddon]

• [x] ❗ I have followed the Contributing to DVCLive guide.

• [x] 📖 If this PR requires documentation updates, I have created a separate PR (or issue, at least) in dvc.org and linked it here.

Thank you for the contribution - we'll try to review it as soon as possible. 🙏

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Part of https://github.com/iterative/studio/issues/8848

Repeat of https://github.com/iterative/dvclive/pull/779 but stripped back to only send subdir.

Depends on https://github.com/iterative/dvc-studio-client/pull/154 / https://github.com/iterative/studio/pull/9173 /https://github.com/iterative/dvc/pull/10303

[codecov-commenter]

Codecov Report

All modified and coverable lines are covered by tests :white_check_mark:

Comparison is base (7305452) 95.51% compared to head (351bbcc) 95.53%.

Additional details and impacted files

```diff @@ Coverage Diff @@ ## main #787 +/- ## ========================================== + Coverage 95.51% 95.53% +0.02% ========================================== Files 55 55 Lines 3542 3559 +17 Branches 317 319 +2 ========================================== + Hits 3383 3400 +17 Misses 111 111 Partials 48 48 ```

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