[BUG] OpenAI BYOK models fail with "Invalid or unsupported model"

# Issue

BYOK OpenAI models fail with “Invalid or unsupported model” in the Playground ChatGPT Client, even when the OpenAI API key is valid.

Free models work correctly. However, selecting any BYOK OpenAI model - including `gpt-4o` and `gpt-5.6-luna` -causes requests to fail with:

> An error occurred: Invalid or unsupported model: gpt-4o

The issue also affects the in-app MCPJam help chat: after switching to a BYOK model, the help chat stops working as well.

# How to reproduce

1. Open the MCPJam Inspector Playground as a local install
2. Select a connected MCP server.
3. Select the ChatGPT Client.
4. Configure a valid OpenAI API key under **Settings → LLM Providers**.
5. Open the model picker and select a BYOK OpenAI model, such as `gpt-5.6-luna`.
6. Send a message in the Playground chat.
7. See the error: `Invalid or unsupported model: gpt-5.6-luna`.
8. Try using the in-app help chat and observe that it also stops working after the BYOK model is selected.

# Expected Behavior

A valid OpenAI API key should allow supported BYOK OpenAI models to be used in the Playground ChatGPT Client.

Selecting a BYOK model should not break the in-app MCPJam help chat. The selected model should be routed correctly using its provider-qualified identifier.