mountain bar diameter is 22.2mm and road bars are 23.8mm, so 1.6mm in dia, 0.8mm off the thickness of the clamp all the way around. Really depends on the brakes as to whether you are likely to have this to ‘spare’. Though TBH brakes are one of the last things I’d want to fail while riding….
As to the initial question – of course there’s no reason you can’t, just it’s likely to result in too upright a position. the newer jones bars which sweep forward a bit would help in this respect. As above, I’d recommend looking at Ragley Luxys or Salsa Woodchippers for bars which give a similar hand position but are more likely to work with CX geometry.