Wow, you really have that little lantern cranked-up! Congrats on getting a cheap one that doesn't leak as they seem to have trouble crimping the fount correctly.
I've been using liquid fueled lanterns and stoves for a long time and have a small collection. I've seen those cooker gadgets for lanterns, but they never really caught-on because the whole setup tends to be a little wobbly with a pan or pot on top.
The heat output will be completely determined by the wick size. I think your lantern probably uses a 3/8" wick whereas a larger lantern like an Air Pilot or Blizzard will use a 7/8" wick and put out more heat. The thing is, by the time you buy another larger lantern it would be cheaper to just purchase a wick stove. The wick stoves available nowadays don't use a flat wick but a matrix of small round wicks almost like mop strings. I have one of these stoves, but never have used it as I prefer using a liquid-fueled Coleman stove as it's less messy and easier to regulate.