Ah yes! Pigsty at Wireless last summer just as it was getting dark was one of my best Morrissey experiences!
To answer the question though, I don't see too much of a similarity in style or content of the two songs - Birthday is much more pacy and dynamic, Pigsty I would categorise with How Soon Is Now, I'm Not Sorry, Southpaw etc. Definitely room both both in the set! Although I'm not opposed to mixing it up either!