In 3.10+, end_lineno and end_offset fields were added to SyntaxError objects and the args tuple.

>>> try: compile('1 1', '', 'single')
... except SyntaxError as e: print(e.args)
... 
('invalid syntax. Perhaps you forgot a comma?', ('', 1, 1, '1 1', 1, 4))

Here, line 1, offset 4 is the first 1-based column not part of the error.

The builtin default sys.excepthook was modified to read and use this new information and mark (end_offset - offset) columns with '^'s.  This default prints what it does to sys.stderr.

The syntax error formatting in the traceback module was not altered.  However, a new method, TracebackException._format_syntax_error, was extracted from TracebackException.format_exception_only so that the former could be overridden by software that simulates interaction.

The printed traceback does not come from line 1348.  That *executes* the user code, but all Exceptions, including SyntaxError, are caught.  If the exception is not expected and the run is not quiet, the exception is output by report_unexpected_exception(), as seen above as 'OUTPUT' and the lines that follows.
https://github.com/python/cpython/blob/5846c9b71ee9277fe866b1bdee4cc6702323fe7e/Lib/doctest.py#L1264

This calls _exception_traceback(exc_info).
https://github.com/python/cpython/blob/5846c9b71ee9277fe866b1bdee4cc6702323fe7e/Lib/doctest.py#L244
This calls traceback.print_exception, which I believe, for syntax errors, ultimately calls TracebackException._format_syntax_error.

I believe that the options for a fix are either
1. Call default sys.excepthook while capturing its output into a StringIO instance.
2. Assuming I am correct above about _format_syntax_error being called, monkeypatch it.  In line 779,
https://github.com/python/cpython/blob/5846c9b71ee9277fe866b1bdee4cc6702323fe7e/Lib/traceback.py#L779
replace '^' with a field with a calculated number of ^s.

I need to do one of these two for IDLE, and may try both.