Framework does not handle refreshed tokens correctly

Initial Checks

• I confirm that I'm using the latest version of MCP Python SDK
• I confirm that I searched for my issue in https://github.com/modelcontextprotocol/python-sdk/issues before opening this issue

Description

I use python sdk with Keycloak as external authz provider. I allowed my mcp client (in my test scenario it is latest VSC) to get AT and RT (using offline_access scope). I have implemented a TokenVerifier that is doing validation of the MCP keycloak JWT. And then in some MCP tool, i use get_access_token() from the framework to use the JWT. I log both values and i can clearly see that it works fine initially, but after the token was refreshed on client-side, i still get the old accesst token from get_access_token() function. Not sure if you intended to support token refresh in a clean way. But IMHO that is a bug that should be fixed soon.

Example Code

Python & MCP Python SDK

1.12.4

Start by reproducing the reported flow with Keycloak, offline_access, a TokenVerifier, and a client-side token refresh. Trace get_access_token() before and after refresh, then verify that its returned JWT reflects the refreshed token rather than the original value.