Dissolve 1 1/2 teaspoons salt in water in a large bowl and stir in lime juice. Soak chicken in the mixture for 10 minutes; rinse and pat chicken dry with paper towels. Pierce the chicken all over with a fork to make many tiny holes.
Stir vinegar, cumin, olive oil, oregano, rosemary, paprika, garlic powder, 1 teaspoon salt, soy sauce, and black pepper in a small bowl; mix in enough beer to form a paste, about 2 tablespoons. Retain remaining beer in the can.
Rub spice paste all over the chicken, beneath the skin, and inside cavity; refrigerate for 1 hour.
Preheat grill for medium heat; a grill thermometer should read 350 degrees F (175 degrees C) with the lid closed.
Sit the chicken upright on the partially-filled can of beer and extend drumsticks to the grill surface to form a tripod shape to hold up the chicken.
Grill chicken with the lid closed until the skin is brown and crisp and an instant-read meat thermometer inserted into the thickest part of a breast, not touching bone, reads 170 degrees F (75 degrees C), about 1 1/2 hours. Let the chicken rest for 5 minutes before slicing.