How to prevent “too broad exception” in this case?

The PEP8 guide you quote suggests that it is okay to use a bare exception in your case provided you are logging the errors. I would think that you should cover as many exceptions as you can/know how to deal with and then log the rest and pass, e.g.

import logging

list_of_functions = [f_a,f_b,f_c]
for current_function in list_of_functions:
    try:
        current_function()
    except KnownException:
        raise
    except Exception as e:
        logging.exception(e)