Ah, yes. Well, what if you increase the sts when you pick them up around the corner (as you suggest)? If the border isn't very deep, the initially picked up sts may be enough to round the corner and keep the border flat. One alternative is to knit the 4 borders separately and straight up two sides (say the north and south sides). Then pick up and knit the border across the other two side (the east and west). That way you wouldn't be turning the corners at all.
As to which increase, I think the KRL (or KLL) is a more organic, less obvious increase but I think that it would help to try each one out on your test swatch.
I'm fascinated by how you'll use this lovely stitch in your project. Do post a photo if you'd like.