[function] `--update-auth-data` doesn't work

**Describe the bug**
We met the issue that need to update token for functions and used the `--update-auth-data`, but it doesn't work.
After kill the function and start it again, the problem solved.

Need to investigate why it doesn't update the token correctly.

**To Reproduce**
Steps to reproduce the behavior:
1. Go to '...'
2. Click on '....'
3. Scroll down to '....'
4. See error

**Expected behavior**
A clear and concise description of what you expected to happen.

**Screenshots**
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**Desktop (please complete the following information):**
- OS: [e.g. iOS]

**Additional context**
Add any other context about the problem here.

Start by reproducing the function token update flow with --update-auth-data and compare it with restarting the function, as described in the issue. Trace the entry point handling that option and verify that the token changes take effect without requiring a function restart.

Written by the indexing model from the issue text.