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.
They also wind-tunnel tested quad Sidewinders on fuselage Y-pylons and found that it was destabilising, needing an even bigger fin extension than that required by Red Top. What they didn't test, as far as I know, was two Sidewinders on the fuselage stations, plus two more on either the overwing pylons or the export underwing ones. I could see that working very nicely, and I have in mind to do a Luftwaffe F.6 with 4 x Sidewinders in this layout some day.
Regarding engines, the J-79 should fit at a very tight squeeze, but the Spey would need so much fuselage re-building that you might as well start from scratch with a new airframe design. The RM-6C from the Draken only managed it's superior afterburner performance because the diameter of the Lightnings' cans is limited by the shape of the fuselage, so you couldn't really use it as a re-fit option. Of course, in whiff world, maybe Rolls-Royce's advanced Rb-106 didn't get cancelled. This was, roughly speaking, an Avon-sized engine with J-79ish technology, so it would be a natural and very powerful upgrade.
There are some interesting advanced Lightning drawings in Chris Gibson's Battle Flight, which have an extended belly pack whose front end overlaps the missle support pack between the fuselage pylons. They have guns in the front, more fuel and a modular weapon pack in the middle, further back than the real-life missile pylons. This pack could have two Red Top pylons, four Sidewinder pylons, three semi-recessed Mk.82 bombs, flip-out 2" RP packs, a semi-recessed Genie, a recce pack or more fuel. IIRC, all of the profiles show it fitted to a two-seat T.55 which was the preferred option for most advanced Lightnings, but I see no reason in principle why the belly pack shouldn't go on the single-seater airframe.
Bear in mind when sending Lightnings to 'Nam that the thing was an absolute pig to work on, since it was essentially a barely-productionised experimental aircraft. Little things like not being able to easily take the wings off to move a damaged aircraft are a right pain in peacetime but can be killer factors when trying to keep a force in operation in an actual war. I can quite believe it getting a good air-to-air score, but I can also see serviceability levels dropping through the floor as running costs go through the roof.