Feeling comfortable on some of the smaller bikes can be difficult at our size but the bikes can handle it no problem. When I was trying out bikes for size I found the pre-08 Ninja 250r, all the super-motos and nearly all of the dual-sports fit without a problem.
For a newbie large rider the super-motos and dual-sports have alot of advantages. They are light, can be dropped with little or no damage and most have modest easily controlled power. The one disadvantage of these bikes for most people is high seat heights but at 6’3″ that won’t be an issue. However super-motos are usually hard to find under $2k. The most SM models have not been around long and they have higher starting prices than other 250s. The highway capable dual-sports are similar except older models are available. The 200 and air-cooled 250s will struggle with freeway speeds. The SM bikes are designed for street use and not really intended for off-road trails though they are more capable off-road then other road bikes.
In sports bikes I found that the new gen Ninja 250r and the Ninja 500r have fairing cutouts that interfere with my knees. They are still rideable bt it seems like the crease near my knee would be uncomfortable very quickly. The pre-08 Ninja 250r does not have this problem and the GS-500F seemed like a great fit.
I ended up with a `06 Ninja 250r. The 250rs are very common and readily available on the used market. You can find nearly new bikes for under $2k. Plus the ninja has plenty of power for road use. Considerably more than most of the other 250s. The riding position is comfortable for new riders also with an upright sitting position.