A "bit of a dick" is literally a subsection of a penis, as for example the foreskin might be. Especially after it is detached from the main.
However the meaning is metaphorical, and is distinct from being an "utter dick" a "complete dick" and variations of the same.
"Being a dick" (utter, complete, etc) is to have a tiny thread of logic to your words or actions, while completely
missing the point, being socially autistic, probably causing offence and maybe causing physical harm to an innocent.
Being a "bit of a dick" is therefore to meet the above criteria, but to be too personally insignificant to really cause more than a
flicker of annoyance or a moment of disdain. Something like being head-butted by an
angry chicken.