I don't know since merely controlling the area is not enough, you'd have to capture (and use) all the stellar energy that comes out of that region. So K4 could be something like "uses the energy of the entire visible universe", but that's a shrinking concept over the time scales in question.

It's not a well-defined scale anyway, since you could probably build a K3-equivalent civilization with only a few star systems using miniature black holes as reactors; conversely, you could easily be a galaxy-spanning civilization but only use a fraction of the total stellar output.

The concept of translating into another universe does seem like an appropriate achievement and next step after K3. Since another universe might have wildly different basic parameters, classical sci-fi portals and such are probably out of the question. You'd have to translate the information stored in the brain into something appropriate that could continue to perform roughly the same functions on the other side.