feat: Set notification options method to configure broadcasted notifications

[joshuazhou744]

Method in FastMCP to configure which change notifications (prompts, resources, tools) the server advertises and broadcasts. Allows developers to selectively enable or disable server-side change notifications announced to clients.

Motivation and Context

Hopefully this change fulfills Issue #1126 with a higher level solution for better user interface.

How Has This Been Tested?

Unit tests added in /tests/server/test_notification_options_propagation.py.

The tests confirm that:

• set_notification_options() correctly sets internal NotificationOptions.
• The configured options propagate to the StreamableHTTPSessionManager and underlying MCP server initialization.
• The configured options propagate through run_stdio_async() to the low-level server's InitializationOptions.

Breaking Changes

None. Existing FastMCP servers will continue to default to no notifications unless explicitly configured.

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

Additional context

Documentation needs to be updated, I am unsure where to update because the docs directory is pretty empty. Maybe a new section in the README?

[joshuazhou744]

If this is redundant to PR #1043 I'll close it. Just saw that PR lol.

[felixweinberger]

If this is redundant to PR #1043 I'll close it. Just saw that PR lol.

At first glance looks like they're complementary, not duplicates - will keep in mind as I come back to #1043.