Generally speaking most Euro karts are a bit lighter, depending on the tubing size and diameter, components can actually ad up a bit too.
I almost wanna say that OTK ( Tony Karts are very light )
Your hight and weight is not that far off from many guys I race with, honestly most guys are at 215, 225, and 230. Many, including myself try to shed 20 to 30 lbs from time to time, with some noticeable success.
Also at 6’2″ tall is not impossible to fit in most karts, I’ve seen bigger guys fitting in no problem, but Margay Brava Limo makes a very suitable kart for taller drivers, and CRG “Big Al ” Is also a very good option, some of the newer Tony Kart EVK are also pretty tall driver friendly.
As for your budget goes, you would have to determine What class and club you wanna race at, and what the spec engine is to be able to race legally.
But if all you want is the good old weekend warrior, then any decent well maintained used kart/engine package will do.
Let us know what your budget is, and engine type to better help you, and steer you in the right direction. Hope this helps. FastFreddy. ;)