How to log actual SMTP Authentication module communication with server?
I guess a simple answer would be to change the SMTP module's debugging output to use the watchdog, in addition to/instead of to the screen. 

Unless the module offers a hook to do that you'll probably need to patch it temporarily, but if you need to debug on the live server you don't really have another option (not an easier one that I can think of, anyway). 

The usual advice would be "get it reproduced on dev, then relax devel permissions temporarily for anon users to test", but of course with emails, getting problems reproduced is often hit and miss between different servers.