I'm not really sure what your question is here (perhaps it got lost during the editing), but let's me assume the only question is contained within the current title "Should I call in sick if I am fatigued?".
Let's review the facts as you have presented them:
- You worked for this company only 1 week before leaving for a 1.5 week vacation
- You are a contractor, not an employee
- You didn't sleep well during vacation, and are fatigued
- You admit that you are not actually sick
- You are strongly considering telling them that you are sick and must stay home for 2 more days
- You suspect that if they were to call you, they would realize that you are not actually sick
I always recommend that people tell the truth. Being honest feels better, doesn't ruin your reputation when the truth is discovered, is easier to remember than lies, and virtually everyone can say it convincingly (while many folks struggle to lie convincingly).
Thus, instead of saying "I am sick. I caught a cold that will require 2 days to heal.", you should either just go to work and catch up on your sleep at some other time, or at least say something like "I didn't get enough rest over the past few days, and think it would be better to take a few days off to recuperate".
While both are honest answers, I would strongly suggest you consider the former - just go to work. Catch up on your sleep on your own time.
I hire a lot of contractors. I expect them to work. If I see signs that they can't (or don't want to) be here and working hard every day, it's very easy for me to replace them. If one of my contractors who had only worked one week decided it was important to extend his/her vacation by 2 days because he/she was tired, I'd almost certainly get a new contractor. And I'm certain I would never bring that contractor into my shop again.
Your situation may be different, and it may not be as dangerous to take off 2 extra days in your shop, so you'll have to assess the specifics of your situation.
But either way, I urge you not to lie. That seldom ends well.