you can build a sbc and mate it too a 2004r in a c3 without much effort if it came with a th350, but the handling concern you point out, between the SBC and BBC is a total non-issue, if you select aluminum heads, water pump and intake manifold on your bbc engine build as the difference in weight between a bbc built that way and a sbc with iron heads and intake is about 30 lb IF THAT MUCH, I know several people that raced BBC C3 corvettes in road racing, in the past, I've even built a few of those corvettes engines and the feeling most guys had was the much greater torque and rapid response far, far out weighed the minimal weight difference, and keep in mind that most of those racing classes had minimal weight limits so the BBC corvette was at some advantage.
lets say either engine in good tune produced 1.4 horsepower per cubic inch,
and
lets say you did not take advantage of aluminum bbc components so you paid a big weight penalty of adding 100 extra pounds ,
and,
you built a sbc with a 377 built for road racing,
(400 block, 350 crank, 6" rods)
the C3 corvette you select after mods for road racing, weighed 3300 lbs
(trust me youll fight to get it that light weight and race ready)
and you compare it to the same car with a 468 cid BBC
(.060 over bore 454)
which are both very common combos.
the
377 x 1.4=528hp divide that power into
3300 lbs=
6.25 lbs per hp
the
468 x 1.4-655 hp,divide that power into
3400 lbs=
5.19 lbs per hp
thats a roughly 19% advantage in power to weight for the BBC powered car,
and thats not pointing out that you could build a 477 (dart 4.5" bore 427 3.76" stroke bbc ) which would make even better power.
lets say we make the cars weigh 8 lbs per cubic inch
377 x 8=3016 lbs
468 x 8=3680 lbs
here you have a
664 weight advantage for the sbc,
and if we use the same power, to calculate
3016 lbs divided by 528=5.71 lbs per hp
3680 lbs divided by 655 hp- 5.61 lbs per hp
this makes the cars about equal because the slight power to weight advantage the bbc has will be negligible to to inertial losses in acceleration and braking,
but your not restricted like that, and its a whole lot easier to get a 3400 lb BBC car built than either a 3000 lb sbc or forcing you to add weight to have a 3680 lb bbc
If you know what your doing, and can build a decent engine and drive train and suspension, with the correct brakes cooling engine,suspension mods and tires,properly set -up a big block corvette with a l88-ls7 can provide impressive race track handling and acceleration and braking.