Where is the fireplace going relative to the structure? Assuming the dude at the end of the video is my height, that puts the top of the chimney at around 66". I wouldn't want that venting onto a pergola, and I definitely wouldn't want it venting into a roofed structure.
Without knowing exactly what you're building and how you're finishing everything it's hard to know how far off your numbers are. I know I couldn't be under $5k for a pergola for that size space, $10k for a roofed structure, but if you're confident, have at it. This isn't even saying you're leaving money on the table, this is worrying that you're hosing yourself.
Take DVS's advice and let the client buy the materials and just charge for your labor. I'd bill weekly, too. When you're dealing with unfamiliar products you're going to hit snags, and when you're dealing with Harry Homeowner stuff like that fireplace you often hit huge snags.