Custom OAuth MCP connector tools not surfaced in mobile chat (works on desktop/web)

### What happened?

Custom remote MCP connector with OAuth completes authentication successfully, and server logs confirm the discovery handshake and tools/list respond with HTTP 200. However, on the iOS app the connector never shows as usable — no tool badge appears, and the UI stays on "Connect." In chat, the assistant reports the connector isn't available as a tool.

This appears related to #472, which describes the same session-restart / discovery-without-invocation pattern on mobile.

Repro steps:
1. Add a custom OAuth MCP connector (also tested with a freshly created connector — same result).
2. Complete OAuth on iOS. Server-side logs confirm successful handshake + tools/list 200.
3. iOS app never shows the tool badge; chat says the connector isn't available as a tool.

Controls:
- A different, smaller connector on the same device/account works normally (tool badge appears, tools invocable in chat).
- The same custom connector works correctly on Claude desktop/web, full tool catalog invocable.
- The connector is visible for artifacts on mobile, but not invocable in conversation — so it's not a total auth failure, just a tool-surfacing issue specific to chat.

Additional test: reduced the connector's tool catalog to a small handful of tools to rule out payload size as a factor. Same failure — still no badge on mobile. Points to a client-side issue in how discovery is wired to invocation for custom (non-directory) connectors, specific to the mobile chat surface.

Already reported via official in-app support as well.

### What did you expect to happen?

Expected the custom connector to behave the same way it does on desktop/web: once OAuth completes and tools/list returns successfully, the tools should be available and invocable in mobile chat, with a visible tool badge — consistent with how other (smaller/directory) connectors already work on the same device and account.

### Steps to reproduce

1. Add a custom OAuth MCP connector on Claude mobile (iOS). Also tested with a freshly created connector name — same result.
2. Complete OAuth on the iPhone. Server-side logs confirm the discovery handshake and tools/list respond with HTTP 200.
3. Return to chat and try to use a tool from the connector.
4. See: no tool badge ever appears on the connector; the app UI stays on "Connect." In chat, the assistant reports the connector isn't available as a tool.

Controls tried:
- A different, smaller connector on the same device/account works normally (badge appears, tools invocable in chat).
- The same custom connector works correctly on Claude desktop/web, with its full tool catalog invocable.
- The connector does show up when building an artifact on mobile, just not as an invocable tool in regular chat.
- Reduced the connector's tool catalog to a small handful of tools to rule out payload size — same failure, still no badge on mobile.

### Area

MCP Connector (adding/managing servers)

### MCP Server (if applicable)

_No response_

### Error messages or logs

```shell

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### Additional context

_No response_

No source files, tests, or entry points are named in the report. Start by reproducing the issue on iOS and compare the mobile chat surface with artifact usage and desktop/web behavior; done means the custom OAuth connector shows a tool badge and its tools can be invoked in mobile chat.

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