The height increase of the fuselage (if you're talking about the pilots' canopy) does not induce a lot of extra drag. The Mirage two-seaters all look like this and that is a highly efficient airframe.
In any case we are talking Mach 1.2 and not Mach 2 with supercruise etc. I reckon the main drag is induced by wing tip/tail tip shapes, and the wing planform.
As you can see, wing was not only made larger, but there is a little twist in the wing-joiner fillet. Also, the tailplanes were made larger and the wingtips and tail's tip were clipped and rounded to reduce drag (a la 80's/90's Russian aircraft like Su-27). The long fins under the exhaust are also gone. I also believe some of the components may be made of GFRP/CFRP to reduce weight.
From left to right: FTC-2000G,JL-10,Yak-130,M-346,T-50,T-X
Now if they could fit a cheap turbofan with FADEC to this design - that'd increase the range quite a bit (at least say 70%). Turbojets are notorious gas-guzzlers, but they are also simpler to maintain and reliable in rough use compared to Turbofans. But Turbofans are more expensive and then we'd have a tough choice to make, a 60's era design with the expense and the relative economics of say, the Gripen. Do we want such a thing?
I think better to just leave it at cheap, somewhat reliable and expendable. That is not a detriment to this machine. It is what it is and is superb as a 'bang-for-the-buck' platform. Kind of like a modern day Chinese T-38 Talon.