Create the GitHub App for board writes and security-alert reads, and store its credentials as org secrets #2544

Problem

Three planned automations write to the org project board or read security alerts, and all are blocked on the same missing credential: GHSA draft cards (#2462), nightly triage (#2463), and the security sweep (#2542). dependabot-alerts.yml already reads secrets.PROJECT_TOKEN, but the secret does not exist, so the daily sweep files issues it cannot board.

The built-in GITHUB_TOKEN cannot hold organization projects: write, so the board write needs a separate identity. The App's spec currently lives inside #2462's body; this issue gives the prerequisite its own card.

What to create

A GitHub App installed on the modelcontextprotocol org, on the inspector repository only. It is only an identity with permissions: no code and no server. Workflows mint a short-lived installation token from it on each run with actions/create-github-app-token.

Grant nothing beyond this list. In particular: no Contents write, no Workflows, no Secrets.

Storage

• The App id and private key go in org Actions secrets limited to the inspector repo (for example INSPECTOR_BOT_APP_ID and INSPECTOR_BOT_PRIVATE_KEY).
• Record the chosen secret names in AGENTS.md, next to the sweep sections that consume them.

Notes

• This is an org-admin act and stays human-gated. An agent cannot create or install the App.
• The fallback is a fine-grained PAT with the same permissions, stored as PROJECT_TOKEN. It is quicker to set up, but it is tied to one person and must be rotated by hand. The App is preferred.

Acceptance

• App created and installed on inspector only, with exactly the permissions above
• App id and private key stored as org Actions secrets limited to inspector
• Secret names recorded in AGENTS.md
• A throwaway workflow_dispatch run mints a token and reads the board, proving the install before the sweeps depend on it

Part of tracker #2543. Every board-writing sub-issue there is blocked on this one.