Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Combine venison and ground beef together in a bowl. Add bourbon; set aside until liquid is absorbed, about 10 minutes. Add bacon and black pepper; mix until thoroughly combined. Form into 4 equal-sized burger patties.
Cook burgers on preheated grill, basting with barbeque sauce throughout cooking, until no longer pink in the center and the juices run clear, about 5 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Top burgers with pepper jack cheese and cook until cheese is melted, about 1 minute.