It prompted Christchurch police to shoot back with: "Come see us and will arrange at no cost."
"If only they were as good at finding me as they were with comebacks," Lake replied.
The police quip has attracted more than 2500 'likes'.
Mr Lake later changed his Facebook profile image to a photoshopped version of the mugshot, altered to look like the Ben Stiller character from the movie Zoolander: