[FEATURE] GitHub Integration connector: expose GitHub Projects v2 (boards) tools

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)

Problem Statement

Running a Claude Routine (cloud) with the GitHub Integration connector connected, there is no way to read or write GitHub Projects v2 board state (items, Status, or other custom fields). Every other GitHub surface works through the same connection: repos, issues, PRs, branches, Actions. But a routine that needs to pick up "the next Todo item from the project board" has no tool that can see the board at all, and has to fall back to weaker signals like issue titles or labels.

Proposed Solution

Add Projects v2 tools to the GitHub Integration connector, at minimum:

• List a project's items with their field values (Status, plus custom fields)
• List a project's field definitions (name, type, options)
• Update an item's field value / move it between Status columns

This needs explicit GraphQL support (projectV2, ProjectV2Item, ProjectV2FieldValue), since Projects v2 has no REST equivalent.

Alternative Solutions

• list_issue_fields (repo-scoped) returns []. It covers a different feature ("Issue Fields," org/repo custom fields on issues), not Projects v2 fields.
• search_issues with a project:owner/N qualifier returns linked issues, but field_values is empty on every item.
• Workaround: tracking phase/workstream via issue labels or milestones instead of the board. Works, but duplicates data that already exists correctly on the board.

Priority

Medium - Would be very helpful

Feature Category

MCP server integration

Use Case Example

A Claude Routine runs on a schedule against a repo with a Projects v2 board organized by Phase and Workstream. It's supposed to pick the highest-priority "Todo" item and implement it. With this feature, it could query the board directly. Today it can't see the board, so it either stalls or risks picking the wrong item from unrelated signals.

Additional Context

Reproduced twice in one session (main session plus a fresh subagent), including a full inventory of about 30 mcp__github__* tools. None are project/board related. The upstream github/github-mcp-server already supports Projects v2, so this may just not be wired up for the hosted Claude Routines connector yet.

<img width="1438" height="576" alt="Image" src="https://github.com/user-attachments/assets/d295d164-1869-4c28-aab8-528991658f44" />

<img width="696" height="263" alt="Image" src="https://github.com/user-attachments/assets/77ae9bb4-0505-47c2-b4d4-450b38d4d76a" />