If You use LED lights, You will be making little heat! Use Incandescents or The quartz work lights - they make a lot of heat! Got any old strings of Christmas lights? Nice distributed heat, no fan needed....
To heat the shack, think of Toaster Oven, Coffee pot, dual use heaters...
A very efficient Induction "burner" can keep cheap iron/steel tea and or coffee pot hot, cooks like a dream with cast iron skillet ( breaded and Fried Fish, anyone?), and would heat the small space nicely, with no flame to steal Oxygen. Bonus: the induction "burners" have multi-level heat control! Using that feature would keep the generator in ECO mode, conserving fuel, increasing runtime, and making less noise!
I bought a WEN brand "2000 Watt" gasoline Inverter generator (Model# 56200i) from Amazon, for an even $400.00. Free shipping because a Prime member. Turns out it is simply a different color case from the Harbor Freight one. Has a Chinese Manufactured motor of YAMAHA design. Good unit, now have circa 50 Hours on it.
Just like the Yamaha, Kawasaki and Honda units, it is NOT 2000 Watts capable when in the "ECO mode", it is only 1600 Watts. But if you run only resistive loads like lights and heater fans with teeny motors, it will run at 3600 RPM (not in ECO mode, noisier and more fuel hungry) and carry very nearly 2000 Watts of load.
Within a dB or two of the Honda figures, weighs very nearly the same (mine is 57 Pounds oiled and fueled), and Cheaper! Yamaha design does NOT have the plastic Cam gear of the Honda, for a few Ounces weight penalty.
That was My happy solution, Your Mileage May Vary.
Regards