fix: add error handling for stderr messages in stdio transport

[Areo-Joe]

• Add catch handler for promise rejection when sending stderr content\n- Log both error and original stderr content when sending fails\n- Improve debugging capabilities for stdio transport errors

Motivation and Context

See #416

How Has This Been Tested?

I have done the reproducing steps mentioned in #416 and the inspector successfully went on to connect and then disconnect and then connect.

Breaking Changes

Nope.

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Documentation update

Checklist

• I have read the MCP Documentation
• My code follows the repository's style guidelines
• New and existing tests pass locally
• I have added appropriate error handling
• I have added or updated documentation as needed

[olaservo]

Hi @Areo-Joe, thanks for the PR. I think that this PR might cover root cause for what you're handling here? -> #380

[Areo-Joe]

Hi @Areo-Joe, thanks for the PR. I think that this PR might cover root cause for what you're handling here? -> #380

yeah it does, great to hear.

[cliffhall]

Hi @Areo-Joe, thanks for the PR. I think that this PR might cover root cause for what you're handling here? -> #380

yeah it does, great to hear.

I'm going to go ahead and close this one, since the previously merged PR covers this. Also, I am not currently able to duplicate it so, we can #416 fixed