"become able" is correct and usable in written English.
It is used to describe the process of acquiring the capacity to do something. For example, "With practice, she became able to speak Spanish fluently."
A stroke victim had become able to feed himself.
You just… adapt - you become able in a different way.
Tens of millions of people have become able to afford expensive cures.
Four days prior to the rash is when most people become able to spread the love.
You become able to do something, and that's almost verging on fatal".
A movie of hyperswarmers starkly illustrates how different they had become, able to fill up the entire dish.
And outside agencies estimate that Iran could become able to assemble a nuclear weapon between 2010 and 2015.
Districts nominate candidates they feel would benefit from this program and will become able and willing administrators.
As in "Jeanne Dielman" we gradually become able to infer a story — this time, of a young woman's growing autonomy and escape from the past.
It is owing to such ties that Australia has become able to leverage influence in the region, whether on human rights or in efforts to fight international crime.
And now we train you on that ball, so that you will become able to ignore that it is a ball".
