"He was always a bit of a lad" sounds British to me, and means (I think) something like "a bit of a tearaway/a bit of a wild one". "a bit of a boy" could simply mean "he's always behaved a bit immaturely." (Suggestions only, from a US EN speaker.)
A 'lad' in this context means a high-spirited, or extrovert sort of man, and probably a bit mischievous and roguish. It can have positive as well as negative connotations.
We wouldn't say 'a bit of a boy'. ''Boy' isn't synonymous with 'lad' here.
I think the closest equivalent in AE might be "a good ol' boy" (ol' = old) which refers -- especially in the American South and West -- to a guy of any age known for being high-spirited and probably very fond of beer.
That's interesting. To my mind, 'a bit of a lad' definitely refers to promiscuity as much as drunkenness and all sorts of unpleasantness. 'Laddish' and 'lad culture'/'laddism' are even worse.
I don't know what 'a bit of a boy' might mean since I've never heard it.