I don't know what kind of debugging info that you expect from WWW::Mechanize? I use WireShark and variants to analyze whether I can get a DNS name resolved, etc.
My internet connect is "fairly reliable", but it will still fail at random unexplainable points - often due to a "barf" at the other end - happens maybe every 3,000 URL's or so. So I just do a re-try which works >99% of the time. Very seldom does one of these things go past one retry and I allow 2 retries.
Logging massive details of all communications in search of an unrepeatable and rare communication failure between you and site X will not help. What are you going to do with that? Sounds like a lot of logging data that will tie your computer into knots while producing no useful actionable information!
If there is some kind of a "bad boy" website (slow, unreliable...whatever), logging that is useful. But you will have that info from the normal functions.