I'm in the same boat you're in. I'm 6'1 and about 280. As the other guys said, useful load is gonna be an issue with most LSA. The exception I know about is the Remos G3 and the GX to a lesser extent. The G3 has over a 700 pound useful load usually. The GX is close to 650 normally. I have sat in a GX. I won't say it was luxurious, but I fit fine. I have flown an Evektor Sportstar, and it is very roomy and comfortable. My instructor and I flew it from northern Ky all the way to sun n fun. Its useful load is usually only around 550 pounds in the earlier SL models you typically find for rent, though, so you'd need to find a very small instructor to make it legal and have enough fuel for anything. I have sat in Flight Designs CTSWs and they are great and have a pretty decent useful load. Forget anything Jabiru. They are super narrow. I couldn't even close the door on a J230. You would probably be fine in a Sportcruiser or a Bristell if you could find one, too.
Good luck with your training. I hope you can find an aircraft near you. I had to give my training up when the only flight school with an LSA within 300 miles shut down.