Hey Matthew - we have a lot of native areas and it is a continual work in progress for us. (FYI I am not a super but I work with mine daily). The first thing we had to do was be sure those areas were not being irrigated. Over the past few years we have replaced hundreds of heads, adding part-circles, or simply turning off some. However, this is not a perfect science, either.
Part of the fix is to also monitor where necessary water is reaching these areas and, it if is impractical to change the head, we have cut back some of the long grass and maintained it as regular rough. And also, be careful if you fertilize through the irrigation system, as it is easy to feed the long grass too.
The question that needs to be addressed is this - can we deal with the fact that some of the rough will be a bit more brown as it nears the native grasses? If the course is adament that green grass must exist right up to the long grass lines, then they need to be prepared to spend the $$$ for it. We have had a company bid to add a line of what are essentially like residential pop-up heads to some of our greens surrounds. We could easily spend $75k to achieve a perfect look - it costs on average $5000 per green complex.
Finally, there were a few areas that looked great, but were simply too much in play to create a problem. We had a spot 200-yards from the first tee that got water and was really thick. We ended up cutting that back and it looks great now, but sometimes you have to sacrifice the native areas. However, we are contantly looking at where we can add the areas too, so in the long-run it works out.