Been doing some thinking during the day ...
The "perimeter rope" (it has a proper seamanly name ...)
I assume that this rope is thinner than a 2 inch diameter hawser (ropes are usually measured by circumference) so it should be in the vicinity of a quarter millimeter (0.25 mm). Trying to sew this with an even thinner thread is bound to fail.
Are the sails cut to exact size yet? Edit: Dumb question, I see them hanging from your "clothes line" in a previous post ...
Maybe this could work:
1. Lay the sail flat, aft side up, on a non-stick surface (waxed paper, thin teflon sheet, glass, you get the general idea).
2. Lay the thread along one edge
3. Use a small paintbrush to pick up thinned (50/50 ??) acrylic lacquer and apply the lacquer to the thread or possibly in the "shadow" under the thread were it lies on the sail cloth. The idea is to glue the thread with lacquer.
4. Wait until the lacquer has set.
5. Repeat from step 2 with the next edge until finished
6. Trim away excess cloth, unless the sail was already cut to the exact size.
Experiment on some scrap sail cloth first. Maybe with less thinned lacquer. If it fails completely then CA might be the only way to go but it is a lot more difficult to work with.
The reason for recommending acrylic lacquer comes from the experience of using floor polish. Johnson Future if you remember it, it was basically an acrylic lacquer, I think it is called Kleer or something nowadays. This is actually a perfectly usable substitute for many of the acrylic lacquers we buy in the hobby stores ...
The cloth I used to polish the floor went stiff when it dried ...
Eyelets: What sizes are we talking about? Less than 2 inches diameter in 1:1 scale? Brass? Use a very fine brush to set a small spot of paint (brass, black, rust ...) where the eyelet should be. Let it dry and make a hole with a very small drill, just large enough for the thread. Stiffen the end of the thread with CA to avoid having to use a needle and insert the thread.
Are we talking eylets as in thimbles on the ends of ropes?
In this case I would paint the inside of the rope loop with CA and when this has hardened and created a solid surface paint it with the desired colour.