Trying to pass magic options through the various layers of shell scripting is entirely the wrong way to go about this on a systemd Linux operating system.
systemd already logs the standard outputs/errors of services that are auto-generated by the "sysv" service generator, as this one is. The "sysv" service generator has made an
exim4.service (somewhere under
/run/systemd) that invokes your
/etc/init.d/exim4 as the service.
There's no delegation going on. Your
rc scripts aren't in charge of the service in the first place. They're simply being run as handy proxies for it.
So what you need to do is look at the already captured log output for the
exim4.service service. This will either be in the journal or in whatever
syslog variant you have arranged to feed off the journal.
For the latter do whatever is appropriate for your
syslog variant. For the former, observe that
systemctl shows you the recent journal entries for the service whenever you run
systemctl status exim4.service
with appropriate privileges (superuser or membership of the
group). You can also view the journal entries for the service since the last bootstrap (that the journal has not already rotated off) with
journalctl -u exim4.service -e -b
exim under proper service management
Ironically, all of that
rc script monstrosity can be replaced with some fairly short
exim4-smtp-submission.socket service and socket units.
Also note that it is a falsehood that exim conflates "foreground" and "debug"/"verbose". Its
-bdf option is explicitly the non-"dæmonizing" version of
-bd, although to invoke it as a per-connection "socket-activated" dæmon (as per the examples in the further reading) where the service management tools handle the listening socket, one would use
-bs rather than