fix(acp): preserve mode across fork

[dawNotPoi]

Summary

• capture the live parent ACP mode when a fork is requested
• reapply a differing mode before selectors become ready and queued prompts run
• ignore stale ancestor state when the current parent does not advertise modes
• add regression coverage for differing, matching, absent, stale, and latest modes

Why

A fork response can reset Claude Code to its default Manual mode while the frontend still caches the parent selection. The next prompt can therefore run under the wrong permission mode. Preserving the backend live mode across the fork keeps the child session behavior consistent with its parent.

Testing

• cargo test --no-default-features --lib fork_mode_inheritance
• cargo test --no-default-features --lib
• cargo clippy --no-default-features --bin codeg-server --lib -- -D warnings
• git diff --check

Fixes #398