"become priority" is correct and can be used in written English.
It can be used to indicate that something has become the most important thing to focus on. For example, "With the deadline so close, finishing the project had to become priority."
Proteases play a key role in cellular biology and have become priority targets for new pharmaceuticals.
The lesson would appear to be that apocalyptic threats are unlikely to become priority concerns.
In an attempt to resolve such problems, these areas have become priority targets for the government's urban rehabilitation programs.
The lesson of recent years would appear to be that apocalyptic threats — when their impacts are relatively far off in the future, difficult to imagine or visualize, and emanate from everyday activities, not an external and hostile source — are not easily acknowledged and are unlikely to become priority concerns for most people.
Thus, the junction points of these transport types become priority areas for SPR location (Niu et al. 2013).
But, naturally, one's own company's business goals become priority and so no player wants to give another even the slightest bit of elbow room, whilst expanding market share.
But many expressed concern that priorities deferred might become priorities abandoned.
The influential Kansei words which affect the consumer satisfaction will become priorities for further design development.
If asylum applicants don't receive relief, they can face deportation and become priorities for removal hence the immigration raids.
Improving access to water and sanitation and reducing corruption within the health sector should become priorities.
Health systems change initiatives in treating tobacco use have increasingly become priorities for national and state level funders.
