Smoking pork shoulder in a little chief
I looked up the Little Chief, and I am of the opinion that it will never achieve a cooking temperature sufficient to cook a pork shoulder. The description suggests it cooks at 165f, which is not far from the 178f you were seeing. That is viable for "cold smoking" foods, but not for safely cooking barbecue of any sort.

You should really be targeting over 200f to do a "low and slow" cook. Personally, I never feel comfortable cooking below 225f.