Expose claude-agent-sdk `taskBudget` option (API-side budget awareness)

## Summary

Please expose [`taskBudget`](https://github.com/anthropics/claude-agent-sdk-typescript/blob/v0.3.144/sdk.d.ts) from `claude-code-action` so reviewers can opt into API-side token-budget awareness. The SDK has supported it since `@anthropic-ai/claude-agent-sdk` v0.2.84 ([CHANGELOG](https://github.com/anthropics/claude-agent-sdk-typescript/blob/main/CHANGELOG.md)); the action pins `0.3.144` as of v1.0.124, so the capability is already in the dependency closure. Today there's no way to set it from the action layer.

## What I'm using `claude-code-action` for

CI code review on every PR in our monorepo (`systemtwosecurity/detectionsai`, ~30 PRs/day). One Opus 4.7 review pass per PR, sticky comment + inline findings.

## Why this matters in practice

We did a 14-day forensic audit of our review runs ([writeup](https://github.com/systemtwosecurity/detectionsai/pull/2657)). Headline number: **37% of completed review runs hit `--max-turns`** despite `--effort xhigh` and risk-tiered turn caps (currently 8/25/45/70 + size adjustments up to 110). The agent has no in-prompt visibility into its remaining budget, so it can't pace verification work — it just hits the cap and exits, often without posting the findings it was about to write up. PR #2495 in our repo (4 successive max_turns runs while the visible sticky still read "No issues found" from an earlier run) is the canonical case.

We've worked around this by manually threading `MAX_TURNS` into the PR Context block of the prompt and adding budget-discipline instructions ("post by turn MAX_TURNS-4"), but the model has no actual budget telemetry. That's exactly what `taskBudget` provides.

## Current state in the action

- `base-action/src/parse-sdk-options.ts` builds an explicit `SdkOptions` (`model`, `maxTurns`, `allowedTools`, `systemPrompt`, `extraArgs`, ...). No `taskBudget` field. Anything in `claude_args` lands in `extraArgs` (CLI passthrough), where the bundled Claude Code CLI 2.1.144 doesn't recognize a `--task-budget` flag and silently ignores it.
- `action.yml` v1.0.124 has no `task_budget` input.
- Grep across the action repo at v1.0.124: zero `taskBudget` / `task-budget` references.

## What I'd like to be able to do

Two options that would unblock us (either is fine):

**Option A — recognized `claude_args` flag.** Have `parse-sdk-options.ts` recognize something like `--task-budget=200000` (or `--task-budget-total=200000` to make the shape obvious) and route it to `sdkOptions.taskBudget = { total: 200000 }`. ~10 LoC.

**Option B — first-class action input.** Add a `task_budget` input to `action.yml`:

```yaml
task_budget:
description: "API-side token budget (total tokens). When set, the model is made aware of its remaining budget. Maps to SDK `taskBudget.total`. See https://github.com/anthropics/claude-agent-sdk-typescript/blob/main/sdk.d.ts (taskBudget option, @alpha)."
required: false
default: ""
```

… wired into `sdkOptions.taskBudget`. A bit more LoC but more discoverable.

Either is fine; both leave room for the API-side option to evolve (it's still `@alpha` in the SDK).

## Workaround alternatives I've considered

- **Fork-and-patch the action.** Doable but creates drift from upstream and we'd lose your release cadence + bug fixes.
- **Bypass the action and call the SDK directly.** Substantial — we'd lose the GH App token handling, sticky-comment plumbing, inline-comment MCP server, etc.

So the cleanest path forward is just to expose the SDK option from the action layer. Happy to send a PR if a maintainer can point at the preferred API shape (Option A vs B).

## References

- SDK type definition: `package/sdk.d.ts:1475-1483` in `@anthropic-ai/claude-agent-sdk@0.3.144`
- SDK CHANGELOG (v0.2.84): "Added `taskBudget` option for API-side token budget awareness, allowing the model to pace tool use within a token limit"
- Beta header: `task-budgets-2026-03-13` (sent automatically by SDK when `taskBudget` is set)

Start with base-action/src/parse-sdk-options.ts and action.yml, then compare the taskBudget definition in package/sdk.d.ts:1475-1483. Decide whether the action should expose a recognized claude_args flag or a first-class task_budget input, wire it to sdkOptions.taskBudget, and verify the configured value reaches the SDK options.

Written by the indexing model from the issue text.