I have used invisble thread before and not had any issues. I have used YLI, Sulky and Superior - I like the cone base that YLI uses - I had troubles with Sulky and the Superior worked great. I used it when I first started quilting as it hides many errors. I have washed and dried those quilt many times in the past 15 years and not had any issues. I also almost never used the invisible in the bobbin - if you do MAKE SURE that you do not wind your bobbin all the way full, can stretch and pop you bobbin out of shape if you do.
The other thing you might want to try is take a few of your scraps, lay them down on a pratice sandwich (backing, batting and then the fabric scraps on top) and try a couple of colors. I am thinking that maybe a dark beige or light to medium tan might work well on them. It really amazes me sometimes how different the thread color (how it sometimes really blends) when it is actually stitched in. I tried this recently with just the scraps left over after cutting and it really worked well - I ended up going with a thread that I never would have thought of.