I've driven a couple of those cars and I've helped build or work on 8 different c4 bbc engine cars, each was totally different , some had mild 427 BBC engine one was a 572 with a centrifugal supercharger , some carbed, most were EFI.
I would suggest the 4l80E transmission and a dana 60 or ford 9" non-independent rear differential and a minimal roll cage to increase chassis stiffness.
keep in mind those cars are now near 20 years old with unknown maintenance and unknown amounts of abuse endured.
You obviously have options, you could purchase the components your want(engine, transmission, differential, oil and trans coolers larger radiator, larger brakes and a simple roll cage and custom assemble the car you want with all new un-abused parts in a couple weeks time if your mechanically inclined and your wallet will handle the strain, and not pay a premium for the lingenfelter name.
the swap may look intimating but if you break the process down to each sub-system and think things through its certainly not something the average mechanic with some experience and a decent place too work, could not accomplish over a couple weeks or months if he has the financial resourced required.
not counting the corvette it self Id expect the whole process would run about $25k-$35K if your just buying everything you might need new, and obviously you have options , but you could certainly reduce that cost with careful shopping , especially if you could do most of the install manual labor yourself and already owed the tools.
related threads (read the sub links)
http://garage.grumpysperformance.com/index.php?threads/the-c4-bbc-engine-swap.151/
http://garage.grumpysperformance.com/index.php?threads/installing-a-big-block-chevy-in-a-c4.6/