[BUG] Claude Code on the web: GitHub write access (push/branch/tree creation) returns 403 "Resource not accessible by integration" despite working read access

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

A Claude Code on the web (Claude Code Remote) session attached to my personal GitHub repo can read/clone the repo fine, but every write operation — git push, creating a branch via the GitHub API, creating a tree/commit via the API — fails with 403 Resource not accessible by integration. This error is specific to GitHub App installation tokens lacking repository access, but I cannot find anywhere in either claude.ai or GitHub's own settings to grant that access.

Environment:

Repo: ac-figa/Social-Analytics (personal account, not an org)
GitHub account ac-figa created 2026-08-19 (same day as this issue)
Claude Code on the web session, environment "Default - full network access"

Steps to reproduce:

Attach a personal GitHub repo to a Claude Code on the web session/environment.
Have the session clone/read the repo — works fine.
Have the session git push a commit, or call the GitHub MCP tools create_branch / push_files — fails every time with 403 Resource not accessible by integration.

Troubleshooting already done (none of it fixed it):

Disconnected and reconnected GitHub under claude.ai Settings → Connectors multiple times. Each reconnect only shows GitHub's OAuth "Authorize" screen — it never shows a repository picker or permissions/installation screen.
Checked github.com/settings/installations ("Installed GitHub Apps") — empty, nothing listed at all.
Checked github.com/settings/connections/applications ("Authorized GitHub Apps") — found an entry called "Claude," but it turned out to be the separate GitHub Actions/PR-bot integration ("Run Claude Code from your GitHub Pull Requests and Issues"), unrelated to Claude Code on the web session access. That page also explicitly states "Claude has not been installed on any accounts you have access to," and offers no "Configure"/install option, only "Revoke access."
Checked every tab in Claude Code on the web's own Settings (General, Account, Privacy, Billing, Usage, Capabilities, Claude Code, Connectors, Plugins, etc.) — no GitHub-specific repository/installation access screen exists anywhere I could find.
Confirmed I'm logged into the correct GitHub account (ac-figa) throughout.

Expected behavior: Either the claude.ai GitHub connector flow should walk through a real GitHub App installation (with a repo picker and Contents: Read-and-write permission grant), or there should be a discoverable settings page where I can grant write access to a specific repo for Claude Code on the web sessions.

Actual behavior: No such flow is ever surfaced. Read access works (implying some token exists), but there is no way I can find to upgrade it to write access.

Impact: Blocks any Claude Code on the web session from pushing committed work back to GitHub for this account/repo.

### What Should Happen?

Either the claude.ai GitHub connector flow should walk through a real GitHub App installation (with a repo picker and Contents: Read-and-write permission grant), or there should be a discoverable settings page where I can grant write access to a specific repo for Claude Code on the web sessions.

### Error Messages/Logs

```shell

```

### Steps to Reproduce

Attach a personal GitHub repo to a Claude Code on the web session/environment.
Have the session clone/read the repo — works fine.
Have the session git push a commit, or call the GitHub MCP tools create_branch / push_files — fails every time with 403 Resource not accessible by integration.

### Claude Model

None

### Is this a regression?

I don't know

### Last Working Version

_No response_

### Claude Code Version

Claude Code on the web session, environment "Default - full network access"

### Platform

Anthropic API

### Operating System

macOS

### Terminal/Shell

Terminal.app (macOS)

### Additional Information

_No response_

Start by reproducing the Claude Code on the web GitHub connector flow with a personal repository, then test read access and the create_branch or push_files MCP operations. Review the connector, GitHub App installation, and repository permission paths described in the issue. Done means a user can discoverably grant repository write access and GitHub write operations no longer return 403.

Written by the indexing model from the issue text.