I probably made some errors in my math, so this isn't going to be exact but you all get the point.
Math looked ok, but you forgot fuel. I used a 737-800 variant. Source is Wikipedia.
A 737-800 weighs 91,108 lb operating empty (which means it has everything needed to fly except fuel) and has a max takeoff weight of 174,200 lb. The difference is 83,092 lbs. Max fuel capacity is 6,875 gallons. A gallon of jet fuel weighs 6.8 lbs so a full fuel load is 46,750lbs leaving 36,342lbs for cargo and people.
So at absolutely full fuel load and maximum seating capacity of 184, the average weight per person + baggage must be under 197.5 lbs.
The max weight will be greatly affected by fuel load, as you are not going to carry 10 hours of fuel for a 3 hour trip. The above is assuming the longest trip possible (max range) of roughly 3,400 miles.