'became' vs. 'become'

Books for foreign learners of English describe sentences like If I ever become a musician, I will grow my hair as the First Conditional and sentences like If I ever became a musician, I would grow my hair as the Second Conditional. The First Conditional predicts a likely event if the condition introduced by if is fulfilled. In the Second Conditional the situation introduced by if is a more remote possibility, one that is unreal, unlikely or untrue.

As long as your verbs agree then they both are technically correct. There may be some linguist psycho-babble about which is which, but they are both mechanically sound.

However, it wouldn't be a bad idea to carefully review the connotations attached to your message.

If I ever become...I will tends to tip towards possibility and away from hypothetical speculation

If I ever became...I would sounds more hypothetical.

I suppose you could say one says maybe someday, the other says if it was possible