Maybe Trapcode Shine’s fractal noise isn’t based on a loopable value (is this something you know should be there?).
Looking at the available properties, Evolution speed is probably the closest equivalent. So you could set keyframes and then add an expression like:
loopOut(“cycle”);
Which would get your keyframes to repeat, or you could go for:
loopOut(“pingpong”);
Which would see the speed bounce between your two keyframe values, or lastly:
loopOut(“continue”);
Which keeps the value increasing, but by moving the keyframes along the timeline lets you set the speed.