There is absolutely no " Safety " issue mixing meats in a Brine or Cure. The typical answer is, " Chicken has Salmonella, Pork has Listeria and Beef has E-coli 0157... They will cross-contaminate. " Who Cares! The Meats are all getting Cooked to a Bacteria killing IT in a 225°+, Smoker. With the exception of Chicken and Pork, we don't care about IT as long as the recommended 165°F and 145°F IT, respectively, are reached. The Beef can be whatever you like as long as it is Intact.
The problem is with Salt Penetration and Brine Time. Chicken only needs 12 to 24 hours in a 2% Salt and 0.25% Cure #1 Brine , by Weight of Meat and Water combined, to reach full penetration and Equilize at 2% Salt and 0.25% Cure #1. Beef and Pork needs 7 Days Per Inch in the same brine to reach full penetration and Equilibrium, AKA, a uniform Salt/Cure concentration. So, in a 2% Brine the Chicken will absorb More than 2% Salt, Unless you Brine it as long as the Red Meat so that the Whole System, Water, Bird and Beef, Equilize at 2% Salt and 0.25% Cure #1. Brining Chicken for say, 7 to 14 Days, will yield Mushy Meat. NO Bueno!
So, unless you are Brine Curing, Chicken Leg Quarters with 1/4" Beef Fajita Strips, overnight...Do the Poultry and Red Meat Separately.
In Contrast, Beef and Pork Equilize at the Same Rate. So if you want to mix Red Meat, have at it! Just make sure they come out of the Brine Cure at the same time, determined by the Thickest piece of meat...JJ