Yes, you will need to extend your third brake light, or go with one that mounts in the center of the spare.
And I too would be a bit worried about using the stock spare holder. It'll start "tincanning" back there, and it'll pull the metal of the tailgate out, then in, then out. Eventually it'll do some damage.
LOTS of spare tire carrier options on the market. And you know I gotta tell you that we make one. :-)