Did you already try setting error handling on a microflow activity?
Update 1 You can try to alter system texts. Tools -> Batch replace -> filter on 'error'. There is an option to change 'An error occured, please contact your system administrator.' I've never tried this myself so can't give any guarantees on the result. :)
Update 2 Use the logging module as an example and create some notification functionality instead of adding log objects to the database.
Another option is to customize the module by adding an after create microflow on the log entity and implement support notification there. You can set the log level to 'error' to solely act on exceptions.