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

[mattseddon]

• [x] ❗ I have followed the Contributing to DVC checklist.

• [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. 🙏

---

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

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

Required by https://github.com/iterative/dvclive/pull/787

Repeat of https://github.com/iterative/dvc/pull/10291 but stripped back to only send subdir.

[codecov[bot]]

Codecov Report

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

Comparison is base (c72e1fb) 90.58% compared to head (687b06d) 90.59%.

Additional details and impacted files

@@           Coverage Diff           @@
##             main   #10303   +/-   ##
=======================================
  Coverage   90.58%   90.59%           
=======================================
  Files         493      493           
  Lines       37757    37774   +17     
  Branches     5454     5460    +6     
=======================================
+ Hits        34204    34221   +17     
  Misses       2919     2919           
  Partials      634      634           

:umbrella: View full report in Codecov by Sentry.
:loudspeaker: Have feedback on the report? Share it here.

[mattseddon]

Currently needs a Studio release

[mattseddon]

@dberenbaum can you PTAL at this now Studio has been released

[dberenbaum]

Should we add a test to https://github.com/iterative/dvc/blob/main/tests/integration/test_studio_live_experiments.py to ensure the subdir is sent to studio?

[mattseddon]

Should we add a test to https://github.com/iterative/dvc/blob/main/tests/integration/test_studio_live_experiments.py to ensure the subdir is sent to studio?

I felt like the complexity that needed to be tested was addressed by test_monorepo_relpath but I have added test_post_to_studio_subdir (Hopefully it passes on Windows)