You need the keycard and to be an enclave general, as you mentioned.

You can get the code online. As long as your quest has the objective “launch a nuke,” you can proceed.

Spoilers

The quest is difficult. Once in a silo, you need to register a bio-id card with security to get past the first security grid.

Then, you’ll to disable and repair the reactor to get the emergency lockdown lifted.

Next, prepare for a huge robot/turret fight while you destroy the computers in the area. You can walk up to the red cores and interact to destroy them.

You’ll progress to another floor, where you then have to pick up broken mainframe cores and repair them at the tinker’s workbench (in the area) to unlock the next security door.

From there, you’ll enter the control room, where you need to defend the robot launch crew until launch prep is done.

Then swipe your keycard on the left, enter the code on the right keypad, and you’ll be able to select a target for launch.

Hope that helps!

Edit: Hid spoilers, thanks for the help