'Working like a dog' is certainly a simile. There is no reason why a simile cannot also be idiomatic, and many expressions are both similes and idioms. 'Flat as a pancake' or 'dead as a dodo' for instance.
However the construction of a simile requires that you put the subject, and then the thing to which it is being compared in that order. (For the comparison, we use 'as' after adjectives, and 'like' after verbs.) For example 'He slept like a log' pairs 'he' and 'a log', with 'like' after 'slept'.
Because 'a flame-like bush' does not use this construction, you have merely a compound adjective rather than a simile.