fix(mcp): spawn bundled MCP servers on pull_request events

[MukundaKatta]

Why

Closes #1245. On pull_request events with the action's bundled MCP servers, the spawned Claude CLI's session init reports mcp_servers: [] and mcp__github_comment__update_claude_comment (and friends) are unavailable, so the tracking comment never updates. The action's tag-mode plumbing is correct: it builds the JSON via prepareMcpConfig, escapes it into --mcp-config '{...}' inside claudeArgs, and parseClaudeArgsToExtraArgs keeps it in extraArgs["mcp-config"] — but the spawned binary doesn't end up registering those servers. Same shape reported in #1191 for an external (Notion) MCP server passed inline.

The cleanest fix is to bypass the brittle stringified-JSON CLI hop: route inline --mcp-config JSON to Options.mcpServers (the SDK's typed in-process registration path used by Anthropic's own SDK demos) and let only file-path values continue through the CLI flag.

What

base-action/src/parse-sdk-options.ts:

• New splitMcpConfigs() returns { inlineServers, filePath } from accumulated --mcp-config values (replaces mergeMcpConfigs()).
• After flag accumulation, lift parsed inline mcpServers onto sdkOptions.mcpServers. Drop extraArgs["mcp-config"] when the only values were inline JSON; keep it when the user also passed a file path.
• Inline-JSON merge order is unchanged (later flags override earlier on key conflict), so user inline JSON still wins over action-prepended bundled servers.
• File-path forwarding is unchanged: the CLI loads it and merges with whatever the SDK already registered.

base-action/test/parse-sdk-options.test.ts: rewrites the mcp-config block to assert the new routing — inline JSON lands on sdkOptions.mcpServers, file paths stay in extraArgs["mcp-config"], mixed inputs split cleanly. Adds a mcpServers === undefined case.

No changes to src/mcp/install-mcp-server.ts, src/modes/tag/index.ts, or src/modes/agent/index.ts — those keep building the same --mcp-config '{...}' string. The fix lives entirely in the parsing layer.

Tested

• bun test base-action/test/parse-sdk-options.test.ts — 28 pass, 0 fail. New 8-test mcp-config routing block covers single inline, multi-inline merge, three-way merge, file-path-only, mixed inline+file, inline-with-other-flags, the bug-report scenario (action + user inline), and the no-mcp-config case.
• Full repo bun test — 665/666 pass. The single failure is validate-env > Direct Anthropic API > should fail when ANTHROPIC_API_KEY is missing in my local shell because CLAUDE_CODE_OAUTH_TOKEN is set in env, unrelated to this change. CI will run with a clean env.

Manual repro for the original bug is the workflow in #1245 (pull_request event, default config, track_progress: true). With this change, the bundled github_comment/github_ci/github_file_ops servers appear in mcp_servers on session init and the tracking comment updates as expected.