"I've been exposed to English as a kid" is a grammatical incongruence as present perfect can only be used when there are no time indications that the action is in the past and no longer ongoing. Since "as a kid" clearly is such an indication, the first sentence is grammatically unaccepatable.
"I'd been exposed to English as a kid" requires some sort of "when something else happened" "before somene did something", where the tense of the "when clause" is simple past. It is used to express anteriority (something takes place before something else)
e.g. "I'd already been exposed to English as a kid before I met my American cousins, so I could understand them even though I could not speak their language competently"
"I was exposed to English as a kid" is what an adult is likely to say to express that when they were children people around them spoke English and adressed them in English.
Simple Past (in the passive form in this instance) to express that "all that" (i.e. the action expressed by the verb) is in the past and it no longer affects the present.