Can I use the old-school gauge cluster with a tach in the middle with an LS swap, and what adapters are needed?

Here is what the documentation says. Each point links to its source; details depend on your exact computer, harness and transmission.

• LS Control System kit: The white bulkhead wire is the tachometer signal with a 2 pulse-per-revolution output. (Chevrolet Performance)
• LS Control System kit: The 2 pulse/rev tach output may correspond to a 4-cylinder setup on some tachometers or transmission controllers. (Chevrolet Performance)
• PSI's white under-dash wire connects to the aftermarket tachometer's green wire. (Performance Systems Integration)
• Gen III LS PCM; LS-series: The number of tachometer output pulses is configurable in the Gen III PCM calibration for compatibility with different tachometers. (LS Engine DIY)
• Chevrolet Performance LS harness: In this Chevrolet Performance harness install, the tach signal came off the bulkhead connector: white wire, location B. (MotorTrend)
• Chevrolet Performance LS harness: The 2 pulse-per-rev tach output requires the tachometer to be set to 4-cylinder. (MotorTrend)

Confirm against the instructions for your exact parts before wiring. More: Tach and speedometer signals in our tuning guide.