My instinct prompts me to say no, since none of the roots for "cope (noun and verb)" are really very similar to "copasetic/copacetic." You've effectively caused me a pain in my heart region, though -- the same pain that happens every time I remember how my "cousin" "lost" my (10 lb.) OED nine years ago.
Briefly -- the roots for "cope" are Greek, Latin, and French (w/the French being just another reference to Latin root) -- all related to the notion of a "head," or "top" -- like a roof, or a hat, or a decision that sums up, or fixes, a problem.
You could say that if things are "copasetic" they're easily "coped" with, but since there's no widely agreed-upon etymology of "copasetic," I can't confirm anything.
I will say that the very first time I heard that word was from Matt Hyland, circa about 1984-5 I adored Matt, and named my cat after him. My cat Matt was all black with a star on his chest, and when Matt went off to Notre Dame we wrote volumes of letters to each other. At the end of each of our letters, we made a line drawing of a cat, seen sitting down from behind with his tail curled up over his head, and then filled in the drawing with black ink. This was our signature to each other.