You ask some good questions. First of all, most of the F-16's fuselage is filled with the jet engine. Rocket engines are much more compact so there would be more room for propellants. The F-16's geometry was optimized for aerial operations with emphasis on high maneuverability. A Black Horse type vehicle would be optimized for high internal volume and low drag.

According to the link, the Black Horse is designed for a useful load of about 33,500 pounds. Some of that would be the pilot, some would be the payload, and the rest would be fuel. My guess is they plan on taking off with a reduced fuel load. That would reduce the landing gear weight, allow for a slower takeoff speed with better abort modes, and have less induced drag during climb.

There is no way a Black Horse style vehicle could carry enough fuel from the ground to fly into orbit. What they're planning to do is take off, climb to altitude, and refuel from a tanker plane. From there, they hope to have enough propellant to fly into orbit.

The refueling idea solves some problems but introduces others. Aerial refueling is delicate work. The pilot has to fly with very precise control and throttle inputs in order to maintain formation with the tanker plane, sometimes for several minutes. Can rocket engines be throttled with that degree of finesse or will they maintain a steady power and use variable drag flaps to maintain formation? How long would it take to fill the Black Horse's fuel tanks? My guess is that it'd take at least a couple minutes to fill the tanks.

Would roughly 30,000 pounds of JP5 and hydrogen peroxide have enough energy to boost the vehicle from roughly 30,000 feet at Mach 0.8 refuelling speed to orbital velicity? I don't have time to run any numbers through the rocket equation right now (nor do I know the Isp of the engines burning that propellant combination) but it just might be possible.