"primary beef" is correct and usable in written English.
You can use it to refer to the main complaint or grievance someone has about a situation. For example, "The primary beef that people have with this policy is that it doesn't sufficiently meet their needs."
The primary beef between them is Oberon's rather arbitrary demand that Titania hand over one of her young attendants.
E-mail messages aside, Mr. Yari's primary beef is with the Producers Guild, not the academy, and what he sees as a flawed procedure for giving credit where credit may or may not be due.
Campusano's primary beef is with Barrick Gold, their subsidiary Nevada, and their Pascua Lama gold mine project in the mountains above the Diaguita people's home in the Huasco Valley of the Andes.
My primary beef with the green economy crowd is not with Van Jones, but to his often overly-ardent disciples that assign way too much credibility to estimates - in my view, guesses - of how many green jobs exist or will be created.
The primary beef-consuming countries of the world (in per capita terms) are Uruguay, Argentina, New Zealand, Australia, and the United States.
Nellore cattle are the primary breed used in beef production in tropical systems.
Beef trim, the primary component of ground beef, is tested by both the Food Safety and Inspection Service FSISS) and meat packers for Escherichia coli O157 H7 under a zero tolerance standard.
A primary concern of beef producers is to improve uniformity of production traits such as carcass weight, fat deposition and carcass composition because the price paid to producers is based on these criteria.
Each chef must incorporate an exotic primary ingredient -- like beef tongue, pigeon or sea urchin -- into every course, from appetizer to dessert.
Western governments have sought to shame and occasionally name the worst culprits, once in a while withholding aid to particular projects.Britain, the second-biggest donor to Kenya after America, recently complained about cash going astray that was meant to beef up primary education.
And yes, there is a primary storyline about the beef between two members, Q-Tip and Phife which is right up there with Paul McCartney and John Lennon or Mick Jagger and Keith Richards apparently.
