Yes… purple wire from bulkhead will trigger the starter solenoid so long as you provide power to the two red wires at the bulkhead.
One of the red wires connects to the positive lug on the starter and the other comes from the… can’t remember, alternator maybe. Both have fusible links, one at the starter and another on the firewall near the brake booster.
You’ll want to retain the brown alternator exciter wire from the bulkhead also
Your PCM probably hooks to its own ECT sensor and you should be able to install a temp sender for your gauge wire (probably green) on the bulkhead.
And a tan wire (probably) for the oil pressure sender. Not sure if the LS PCM’s need an oil pressure sender of their own, if so you’ll need to figure out how to run a secondary sender for your gauge.
If the PCM deals with it’s own coil wiring you can eliminate the pink (probably) wire that powers the ignition coil.
My wire colors are based from my ‘80 but GM is pretty consistent with keeping wire colors the same.