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)

You can just put a comment like except Exception as error: # pylint: disable=broad-except that's worked for me actually. I hope it could be work for you.

Use this to cheat PEP8:

try:
    """code"""

except (Exception,): 
    pass

I think in some rare cases catching general exception is just justified and there is a way to trick PEP8 inspection:

list_of_functions = [f_a,f_b,f_c]
for current_function in list_of_functions:
try:
    current_function()
except (ValueError, Exception):
    print(traceback.format_exc())

You can replace ValueError by any other. It works for me (at least in PyCharm).