[FEATURE] Make the auto-mode permission classifier timeout configurable

Problem Statement

In auto permission mode, every tool call passes through the small-model safety classifier with hardcoded timeout budgets (observed in v2.1.258: 60 s base, scaled up with context size, ~120 s cap per stage, up to ~5 min including retries). When the classifier times out, the call fails closed — the model receives a "temporarily unavailable, retry later" denial, which stalls the agentic loop.

None of this is configurable: the CLAUDE_CODE_AUTO_MODE_* env vars expose the model, temperature, retry counts, etc., but no timeout; the server-side config gate likewise has no timeout field.

Two failure shapes in practice:

1. High-latency / unstable network to the API: the fixed budget is either too short (repeated fail-closed denials with backoff) or needlessly long.
2. Unattended runs (scheduled tasks, CI): a classifier outage wedges the session — see #85820 for permission requests wedging scheduled sessions with no timeout.

Proposed Solution

Expose the classifier timeout as a setting, e.g.:

• settings.json: "permissions": { "autoModeClassifierTimeoutMs": ... }
• or an env var like CLAUDE_CODE_AUTO_MODE_TIMEOUT_MS

Ideally two knobs (overall budget + single-attempt timeout), with defaults unchanged from today's behavior.

Alternative Solutions

• Switch out of auto mode (e.g. bypassPermissions / acceptEdits) — loses the safety property that motivated using auto mode in the first place.
• Manually re-running blocked actions — not viable for unattended sessions.

Priority

Medium - Would be very helpful

Feature Category

Configuration and settings

Use Case Example

1. I run Claude Code in auto mode for overnight scheduled tasks on a machine with a high-latency route to the API.
2. The classifier occasionally exceeds its timeout budget and the tool call is denied fail-closed ("temporarily unavailable").
3. With this feature, I could raise the budget to match my network conditions — or lower it so failures surface quickly and the run can fall back to a different permission strategy.
4. Today there is no knob; the only options are retry-by-luck or abandoning auto mode entirely.

Additional Context

• Observed behavior (v2.1.258, from the installed binary): token-scaled deadline min(120 s, 60 s + 10 s per 50k tokens above 50k) for the fast stage, fixed 120 s for the thinking stage, ~5 min overall retry ceiling; timeouts surface as automode-unavailable denials (fail-closed); read-only operations bypass the classifier.
• Related: #69002 and #74311 (classifier model/effort configurability — the same gap for timeouts), #86339 (classifier timeout never firing), #86926 (distinguishing timeout from denied).

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)