[Bug]: Custom Antigravity binary path is per-profile and must be duplicated across all profiles

Before submitting

• I searched existing issues and did not find a duplicate.
• I included enough detail to reproduce or investigate the problem.

Area

apps/desktop

Steps to reproduce

1. Configure two or more Antigravity profiles in T3 Code (e.g., separate Google accounts for personal and work).
2. Set a custom executable path under Profile 1 (binaryPath).
3. Switch to Profile 2 under Settings -> Providers -> Antigravity.
4. Notice that Profile 2 does not inherit or share the custom executable path and instead defaults to the managed path in ~/.t3/tools/antigravity-acp/....
5. Whenever a custom binary path is changed or updated, it must be manually updated across every configured profile.

Expected behavior

There should be a shared/global Antigravity binary path setting (or an option to "Use shared binary across all profiles"), so multiple profiles can use the same local installation without redundant manual entry or re-downloading.

Actual behavior

binaryPath is strictly scoped to each individual profile draft. Users with multiple profiles must copy-paste the path into each profile individually, and changes must be manually synced across all profiles.

Impact

Minor bug or occasional failure

Version or commit

0.0.42 (and main)

Environment

Windows 11 x64, T3 Code Desktop 0.0.42

Logs or stack traces

# No crash trace; configuration is isolated per-profile in settings store

Workaround

Manually copy-paste the custom binary path into the settings of every configured Antigravity profile.

Suggested Fix

Allow defining a top-level default binaryPath on the Antigravity provider that individual profiles inherit unless explicitly overridden.

Start in apps/desktop by tracing the Antigravity provider settings and the settings store entries for the per-profile binaryPath. Check how provider drafts are persisted and switched between profiles, then define how a shared path and explicit per-profile override should behave. Done means changing one shared path works across profiles without redundant updates, with coverage for inherited and overridden values.