Yes, trim it off.
The skin can be used to make chicharrones if you want to do that.
Although I haven't done it with a picnic alone but I like what Rich is suggesting. I do this when cooking a whole hog. I crisp the skin on high heat at the end and add it back to the PP.My opinion is leave it on . I remove it after the smoke . Scrape any un-rendered fat on the underside , chop it up and add it back in the pulled meat . What I don't eat , I add back to the meat anyway .