Do not worry about painting it. Here is the problem. No matter what material you use, it is likely to be larger than scale. So black or white colors, the most visible, make it look larger than it is. A medium gray or tan is much better. You want it barely visible. Look at old pictures of these ships, the antennas and rigging are barely visible or maybe don't show up at all.
Rigging that really stands out visually marks it as a model right away.
To reduce visibility further, I use the nylon monofilament stuff, which is translucent. However, this stuff is very hard to work with and I do not recommend it to someone who has not already done a lot of rigging. Instead, find the finest thread you can find, in a subdued color (gray or tan). Finest readily available is about 7 mil. Don't go any larger than that. Model Expo is about the best source to mail order fine thread.
BTW, I recently bought the Konig and the PE set to go with it. However, it will be awhile before I finish my Reale de France, so it will be awhile before I get to the Konig.