cleanup(bigquery): make error mod pub; cleanup allows

[dbolduc]

I need to make the error mod public, and then I rearranged some things.

Fixes #5664

[gemini-code-assist]

Code Review

This pull request reorganizes modules in lib.rs, exposing error as a public module, and renames the handle field in Runner to _handle to suppress dead code warnings. Feedback suggests using #[allow(dead_code)] on the handle field instead of renaming it, which avoids propagating the rename to all destructuring patterns in the test suite.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 96.07%. Comparing base (0dac6ea) to head (2ab7637).

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #6252   +/-   ##
=======================================
  Coverage   96.07%   96.07%           
=======================================
  Files         275      275           
  Lines       68829    68829           
=======================================
+ Hits        66126    66129    +3     
+ Misses       2703     2700    -3     

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