For better cornering out of muddy or sandy corners, look for bigger lugs on the tread shoulders. For example, here are two comparably priced Bontrager gravel tires:
The deeper, more widely spaced lugs on the bottom tire make it better suited for rough, muddy, or sandy conditions, at the expense of speed and handling on hard, smooth surfaces. I can't tell you what to buy. All I can tell you is that a fat MTB tire won't fit in your frame. FYI, an inch equals 25.4 millimeters, so a 2" MTB tire is about 51 mm. Unless your current tires leave a gap of over 1/2" to the frame all around, they won't fit.