Ability to move operations outside the galaxy is probably available at Kardashev II.

Just moving out of the solar system would be pointless, except to obtain a ready supply of cold. We may expect cold to remain essential to anybody still bound by laws of thermodynamics.

The whole "Kardashev" business, anyway, comes from an extremely primitive outlook. Once a culture gets a handle on mass/energy conversion, relying on stars to provide process energy gets left behind, and other resource limits become important instead. Safety from nearby magnetar quakes and merging black holes, for example.