Remove Mutex from bitwarden-json

[dani-garcia]

🎟️ Tracking

With the mutability changes to the client, we don't need a Mutex in bitwarden-json anymore either, which means we can remove the async-lock dependency as well.

⏰ Reminders before review

• Contributor guidelines followed
• All formatters and local linters executed and passed
• Written new unit and / or integration tests where applicable
• Protected functional changes with optionality (feature flags)
• Used internationalization (i18n) for all UI strings
• CI builds passed
• Communicated to DevOps any deployment requirements
• Updated any necessary documentation (Confluence, contributing docs) or informed the documentation team

🦮 Reviewer guidelines

• 👍 (:+1:) or similar for great changes
• 📝 (:memo:) or ℹ️ (:information_source:) for notes or general info
• ❓ (:question:) for questions
• 🤔 (:thinking:) or 💭 (:thought_balloon:) for more open inquiry that's not quite a confirmed issue and could potentially benefit from discussion
• 🎨 (:art:) for suggestions / improvements
• ❌ (:x:) or ⚠️ (:warning:) for more significant problems or concerns needing attention
• 🌱 (:seedling:) or ♻️ (:recycle:) for future improvements or indications of technical debt
• ⛏ (:pick:) for minor or nitpick changes

[github-actions[bot]]

Checkmarx One – Scan Summary & Details – 651fea15-cd1b-4958-b7f6-c8ff5b6b1404

No New Or Fixed Issues Found

[codecov[bot]]

Codecov Report

Attention: Patch coverage is 18.75000% with 26 lines in your changes missing coverage. Please review.

Project coverage is 60.09%. Comparing base (07abbc7) to head (6dff880).

Files	Patch %	Lines
...s/bitwarden/src/secrets_manager/client_projects.rs	0.00%	6 Missing :warning:
...es/bitwarden/src/secrets_manager/client_secrets.rs	33.33%	6 Missing :warning:
crates/bitwarden-json/src/client.rs	0.00%	2 Missing :warning:
...s/bitwarden/src/secrets_manager/projects/create.rs	0.00%	1 Missing :warning:
...s/bitwarden/src/secrets_manager/projects/delete.rs	0.00%	1 Missing :warning:
...ates/bitwarden/src/secrets_manager/projects/get.rs	0.00%	1 Missing :warning:
...tes/bitwarden/src/secrets_manager/projects/list.rs	0.00%	1 Missing :warning:
...s/bitwarden/src/secrets_manager/projects/update.rs	0.00%	1 Missing :warning:
...es/bitwarden/src/secrets_manager/secrets/create.rs	0.00%	1 Missing :warning:
...es/bitwarden/src/secrets_manager/secrets/delete.rs	0.00%	1 Missing :warning:
... and 5 more

Additional details and impacted files

```diff @@ Coverage Diff @@ ## main #839 +/- ## ========================================== + Coverage 60.07% 60.09% +0.01% ========================================== Files 186 186 Lines 12246 12247 +1 ========================================== + Hits 7357 7360 +3 + Misses 4889 4887 -2 ```

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