Ok, one thing nobody has really answered here was the question of what works best with a KA. The reason why they probably didn't answer that part of your question is because it's not really the determining factor. While ever part of an induction system (including an engine) plays a part, the real consideration goes with which turbo, intercooler core, and boost level you are running. Of course, a KA24DE has a higher compression ratio than, say, a CA18DET, and that will also promote a faster spool and more power at the same psi than a CA18DET, but you're asking about turbo choices and intercoolers, so let me stick to that.
Like any setup, the intercooler is just one jigsaw puzzle piece of a larger integrated system that includes fuel. If you pick a T3/T4, it's overkill on a moderately improved fuel system, because you will never utilize the entire boost map, and will add more lag from the larger wheel. A T3/T4 however can flow more air in a denser, colder temp than a smaller turbo, which has to spin more times per second to make the same boost levels.... that extra spin creates more heat, which means more expansion during the air's travel through the intercooler piping and intercooler itself.
But let's stick with what you're trying to figure out here. Your goal is 250hp. The two things that govern choice of turbo are overall budget and power goals. You gave us one of the two, so we can work with that. 250hp on a KA24DE standard longblock is attainable with a standard T25G from an SR20DET, but requires a lot more supporting mods, like full 3" exhaust, big intercooler, lots of boost... The T25G can do it, but it's spinning like crazy to do so, so the air temps leaving the compressor are kinda hot... the intercooler does change that, BUT...
A T28 would be better suited for this. You would be able to make the necessary boost levels without spinning the turbo as fast, which means less heat in the outlet air... that means you can go with a smaller intercooler, because the air isn't as hot to begin with. A T28 will still hit pretty damn hard. You could probably get away with a medium-sized core, instead of a large one.
My suggestion for your power range is a T28 turbocharger, with an intercooler like this:



That bumper was modified further to make it fit behind the stock crash bar, but this was to give you an idea of the ideal size for your desired power goal. This is actually good for more than 300hp, given you get air to it.
After 250-275hp, the KA24DE stock longblock is going to begin to have issues, and you'd really need to build the bottom end at least, with some thick rods and forged, lower compression pistons. The above intercooler will take you higher, but once you get into seriously BIG horsepower, you're going to want to go to something more like this:



That's Bart's car... Bart is primarily a drag racer, so a bigger IC is better for him... and wold also be better for any car at over 350hp, regardless of the type of racing you do.
Here's another large-opening hacked version of the bumper, a bit more modest compared to Bart's car:


You don't necessarily have to hack out half your bumper and all of your crash bar. You could do what I'm going to do, which is this:
(photo is of Keith's car, who's IC is more like the first one above, but assume it's like the second one above based on my photochop)
Factory crash bar:

Cut out...

...and plate or brace welded in:

Blocked:

Unblocked:

You could leave it like that, or paint that top part of the IC black, or all of the IC black, or do what I'd do, and use a mesh kit on the new upper airdam.
Doing this will also send more air through the top of the IC to the top of the radiator, so that's a nice bonus.
Somewhere I found someone on (I think) nicoforums who has an S12 and did exactly this... I have the pics somewhere on my photobucket or hard drive, as soon as I find them I'll edit this post and put them here.
Edit: found the pics...
