They looked at integrating Sparrow on Lightnings, but it proved impossible with 1960s technology to add the CW illumination fuction to the AI.23 radar. It's not so much the diameter of the dish, more the fact that the whole radar is very tightly packed into the intake bullet and there's just not room for extra black boxes or slightly bigger versions of existing ones. The only way to do it would be to use a completely new radar but that's an order of magnitude more expensive.
Reading Battle Flight, it says that in the end Red Top did what Sparrow did so they didn't bother with all the extra work they would have had to do. I've always wondered what an F-4 would have done with Red Tops
If that's the case just go for extra Red Top on the wing stations, a graphic I saw indicates the rocket pod load out was up to two on each of the four over and under wing stations so up to 8 Red Top plus 6 ADEN for my Whiff
I'm pretty sure you could get two Red Tops on the overwing pylons, but four might be excessively draggy due to the size of the T-shaped pylon you'd need to get them fin clearance.
Red Tops on the outboard underwing ones would have me worrying about aeroelastic loads. The stores cleared for them (bombs and rocket pods) were short and clean, whereas a missile is long and has fins that generate their own air loads. You might get away with a Sidewinder there, since it's not very heavy even if it does start "nodding", but you'd still want to tunnel test it
very carefully....
Now with the Lightning serving into the 90s or 2000s then we have AMRAAM and ASRAAM and no need for a CW illuminator
True, but you still need to generate target data to feed to the missile's INS before launch and datalink to it after launch, and that still needs a bunch of electronics and computing power on the fighter. Remember, it took two rounds of updates to get the Tornado F.3 full AMRAAM capability. This doesn't mean it's undoable on the Frightening, just that it's a non-trival and potentially expensive exercise. The Sea Harrier F/A.2 managed it and SHAR's arn't exactly roomy either, but then it did have a complete new cutting-edge radar with a lot of the neccessary functions built into it. Just as on the Sea Harrier (but for different reasons) the Lightning
might have to give up it's guns to get AMRAAMs.
On the other hand, I could see ASRAAM going very well on the Lightning. The missile is small and clean so you could possibly carry four on fuselage Y-pylons without adding too much side area, and the missile isn't very demanding, so the neccessary electronics might fit in the missile support pack area. ASRAAM is a "long-range short-range" missile with a potential flight range of 10 miles (i.e. further than Red Top) and capable of head-on lock-ups, so even if you can't generate INS data to launch it before lock-on, it still provides a formidable capability in lock-on-before-launch mode.