I have the same issue, this happens sporadic, but it’s like 50/50 it will work inside ChatGPT or not.
Reported the bug already, but no response for about a month, since I got the problem the first time.
Usually to get it working again, I’m refreshing the connector and closing the webpage completely. Problem is worse on GPT-5.2 - it often just refuses to use MCP, like connector does not exist. With GPT-5 and GPT-5.1 tools are invoked, but no actual requests are sent to the MCP Server. At the time tool invocation not woring; triggered connector refresh operation (Init/New Session/List Tools/Read Resource) are sent to the server and processed correctly.
So MCP Connector init/auth always working, tool invocations are randomly failing.