Don't know what's going to fit your price range, but at 6'7", somewhere north of 300 lbs, and a size 15 shoe, the one CUT I've found that fit me with no modification was the New Holland TC-45DA. It has the twin peddle HST setup. Its FEL joystick control worked very well, even with my tall legs. Its tilt/telescope steering wheel set up was also very helpful. The seat went back far enough to be usable, but I would want to move it a couple inches further back for ultimate comfort (the seat track design would make this an easy task). At "only" 6'4" /forums/images/graemlins/wink.gif, you probably wouldn't need to do this.
The John Deere 4020 series does not fit quite as good as the New Holland, but not too bad either. However, its FEL joystick control lever, which is a long lever with a crook in the top to bring the lever closer to the operator would need to be modified to be a little taller to work with my very tall legs (the crook hits my right leg, but goes over the top of normal height legs). Also, the way the seat track mechanism is designed on the 4020 series lends itself to a very easy modification to gain many more inches of rear travel, which I would need to do to make it truly comfortable. As with the New Holland, you may not need to deal with the FEL joystick issue I do.
I've also sat on Kubota Grand-L tractors. These did not fit nearly as well as the New Holland or JD. Their seat track mechanism, location, and the built in toolbox behind the seat would make modifying it for more rearward travel difficult at best. Also, the Kubota treadle peddle control for their HST is absolutely unusable for me, my long legs, and my size 15 feet. It was abysmal! However, the GST tranny was marginally usable, but the seat would need to be moved way back to make it comfortable, which might not be possible.
To address your "can the seat be modified" question... As someone who has had to deal with trying to make all kinds of vehicles fit me over the years, the easiest seat track mechanisms to modify are where the seat attaches (with removable bolts!) to the seat track in a flat plain. This allows you to have some custom brackets fabricated that bolt to the seat track in the position where the seat would normally bolt and then allow you to attach the seat to the brackets 4 to 6" back from where it would normally attach. 'Course, this assumes you have the room behind the seat to extend it backwards (a big issue with the Kubota).
Hope this helps.