Your 390mm result is not correct if the compact’s 1365mm figure is a 35mm-equivalent focal length, which it almost certainly is.

Equivalent focal length is: true focal length × crop factor

So if 1365mm is the compact’s full-frame-equivalent reach, the actual lens focal length is about: 1365 ÷ 5.6 ≈ 244mm

To get the same framing on a 1.6× crop camera, you need a lens whose equivalent focal length is also about 1365mm: 1365 ÷ 1.6 ≈ 853mm

So you’d need roughly an 850mm lens on the 1.6× camera for the same angle of view, not 390mm.

The key concept is angle of view: the combination of focal length and sensor size determines how much of the scene fits in the frame. “Reach” is really just a narrower angle of view.

In practice, lenses that long are large and expensive. A more realistic option mentioned in the answers is something like a 400mm lens with a 2× teleconverter (about 800mm actual, or 1280mm equivalent on 1.6×), or a long zoom such as a 150–600mm, though it still won’t quite match 1365mm equivalent.