As has been posted many times, in and of itself, a turbo swap with a bigger unit and running more boost will result in a negligable increase in HP in a Diesel if it's not accompanied by an increase in fuelling. So, I'm left questioning the claims from your source of a 15-20 HP gain from a turbo change alone, but I would say, "no."
Neglecting the effects of the intercooler, a reduction of IAT at a constant boost pressure can only occur if there is an increase in the compressor efficiency at that particular operating point. While this is not clear-cut, I do recall seeing a map of a K03 and it seems to be near it's airflow limits at 19 PSI boost and redline on a TDI, with efficiency only in the 60-65% range. A larger, higher flow turbo will certainly be more efficient in this area (near redline), but the location of the efficiency island is not affected significantly and the peak efficiency also doesn't change much (between 72-74%). I assume most of your time is not spent at max load and redline, so the real-world benefits of a bigger turbo will not be frequently noticeable.
Also bear in mind that the K04 would be from a gasser application, and that means a number of things: trims and A/R ratios that are optimized for higher flow / lower boost, and higher energy exhaust (higher EGT in a gasser). Therefore the A/R will tend to be a higher number, which is good to make more top-end power, but more laggy on the bottom-end.
Also, unless you're getting a hybrid turbo, the centre section of the K04, like most gasser turbos, would be water-cooled, which means necessitating plumbing a coolant supply- and return line.