"helping me become who I am," as a contraction of "helping me to become who I am," is the proper logical grammatical usage. It is a sentence full of meaning. A mentor, for example, could be helping me in my effort to be who I really am, rather than what I pretend to be.
"Helping me becomes who I am" jars, not because it offends rules of grammar but because it is nonsensical. In this sentence, "Helping me" is the subject, followed by a third person singular of the verb "become" and making "who I am" the predicate. The sentence is grammatical but makes no sense, because the act of "Helping me" can never be equated to "who I am." "Helping me" and "who I am," as two stand alones," are not compatible.
When "becomes" is used, the entire expression is to be interpreted as the act of "Helping me" (in the sense of my getting help) somehow getting changed (i.e. "becomes") and the result of the change is "who I am", which makes no sense at all.
Compare "Seeing poorly is what ails me." That is both grammatical and meaningful, because "seeing poorly" and "what ails me" are compatible. We could even say "Seeing poorly becomes what really ails me". In other words, if I keep on using my poor eyesight to read, say, small characters, that exercise will turn into something that really ails me.
This analysis requires no research, merely the application of well established grammar rules and cognition of what makes sense and what does not.