IIRC part of the issue in putting anything other than Firestreak/Redtop on the beast was the missile cooling (ammonia?).
The Firestreak used ammonia for cooling, the Redtop used 12,000 psi pure air. Cooling was only for the IR seeker heads, the electronics used (IIRC) Arklone or Trike. The reason for all those 40-gallon drums of water on the flight line was in case a liney copped for a face-full of liquid ammonia - your mates were supposed to dump you head first into a water barrel. Obviously, in some cases all you had to do was give a liney a funny look and you'd end up in a barrel...
I can't think of a reason why Sidewinders couldn't have been fitted - all that gubbins in the missile pack could have been replaced with ballast, the Sidewinder doesn't need much support from the airframe, unlike Firestreak/Redtop. Given the respective weights of Redtop vs Sidewinder you could have got 2 Sidewinder on each pylon. I was on 5 Sqn at the time of the Falklands war and there was talk of underwing hardpoints for Sidewinder carriage, though nothing came of that. Might have got away with overwing Sidewinders a la Jaguar, though.
Liquid ammonia, liquid oxygen, AVPIN, intake checks, it's a wonder any of us groundcrew survived!