How about preventing this function from being bound as a method from the very beginning? You can wrap the function you're testing in a staticmethod.
This signals to Python that the function doesn't receive the instance as its first argument. As a result, you can call it directly, which simplifies your helper method and makes the type Callable perfectly accurate. Here is a example:

from typing import Callable, Any, Optional
import sys

# Assume this is your external library in mymodule.py
class mymodule:
    @staticmethod
    def myfunction(s: str) -> str:
        return f"processed: {s}"

# Your test framework
class MyTestCase:
    function_under_test: Optional[Callable[[str], Any]] = None

    def assert_something(self, input_str: str, expected_result: Any) -> None:
        if self.function_under_test is None:
            raise AssertionError(
                "To use this helper method, you must set the function_under_test "
                "class variable within your test class."
            )

        result = self.function_under_test(input_str)
        assert result == expected_result
        print(f"Assertion passed for input '{input_str}'!")


class FunctionATest(MyTestCase):
    function_under_test = staticmethod(mymodule.myfunction)

    def test_whatever(self) -> None:
        self.assert_something("foo bar baz", "processed: foo bar baz")