Haha, actually I figured it out like 10 minutes after I posted that... Just had one thing wrong. We were both trying similar approaches.
If the angle of the Gamepad analog axis is outside the deadzone, store that angle in a variable on the object. Every Tick, rotate the object toward that variable, instead of the active angle. This way the variable will only be stored if its outside the dead zone, and not return to zero.
I'm sure there's a more optimized solution, but this is working at the moment.