"become a rival" is a correct and usable phrase in written English.
For example, you could write, "My brother was determined to become a rival for my father's business empire."
Allowing the people to become a rival to the government might bring down the system.
He tried and failed to become a rival to eBay in auctions.
DC Air, by aligning with American, would become a rival of United's rather than a partner.
The trading currency used by a huge barter exchange could become a rival to many weaker national currencies.
The initiative process, originally meant as a safety valve, has in reality become a rival to the legislature.
This proposal was blocked by Lee Kuan Yu of Singapore, who feared it might become a "rival entrepreneurial city".
This happened just before his father became a rival.
Invited into the National Woman Suffrage Association by Susan B. Anthony, Woodhull soon became a rival for leadership.
Needless to say, that idea makes politicians nervous in case the office becomes a rival seat of power.
268 Marcus Cassianius Latinius Postumus, (died 268), Roman general who, by setting himself up as an independent emperor in Gaul about 258 268 became a rival to the emperor Gallienus.
It never became a rival to IBM as French politicians hoped, and eventually exited the computer business altogether.Plus ça change, plus c'est la meme chose.
