I'm riding on the coattails of Mick O'Hea here. You are ...
... a beetle.
My bark is worse than a bite from me.
An allusion to bark beetles. Some beetles bite, but their bite isn't dangerous.
Cleopatra has nothing to do with me!
Beetles are of the order Coleoptera, which sounds like – but has nothing to do with – Cleopatra.
Some say I'm from Germany, maybe they know.
The Volkswagen Beetle is a classic car from Germany.
Some say I'm from England, A+ if it's so.
The Beatles are from England. The "A+" indicates that they are written with an a, I guess.
Some say that I'm crapulous, a literal shame.
An allusion to dung beeltes. "Crapulous" is related to drink, though, not to crap, and it's a "literal shame" to confuse the two (OP).
Look thrice to the heavens and tell me my name.
In a film from 1988, the ghost Betelgeuse can be summoned by calling his name three times. The name sounds like Beetlejuice, which is also the name of the film (OP).
Hint: I am the lesser (and the greater) of two evils.
A pun on "the lesser and the greater of two weevils" (from OP).
Hint #2: I am neither a historic, giant adversary, nor am I a rhinoceros. Or am i?
No, you are not, but there are Goliath beetles and rhinoceros beetles.
Hint #3: This puzzle was created under Puzzling License # LMW-28IF
Mick O'Hea found out that this is the licence number of a VW Beetle on the cover of The Beatles' Abbey Road album.
Title: I could tell you some war stories
Beeltle Bailey is a long-running cartoon about life in an army camp (OP).