Computation of stagnation temperature behind the shockwave in front of a hypersonic projectile is Really, Really Hard, so I'm not going to attempt to do that right now.
Various Space Gun projects have arisen over the years, and the general consensus seems to be that a gun shooting at escape velocity (>11km/s) at low altitude is basically impractical, though I haven't found anyone who has shown their work here. Your 50000fps estimate is equivalent to about 17km/s, and is therefore almost certainly too high.
The SHARP light gas gun was intended to reach muzzle velocities in the region of 8km/s. I assume that the project scientists and engineers believed this to be practical, but the project was more or less discontinued. The Quicklaunch light gas gun was intended to fire somewhat larger projectiles at more like 6km/s. The StarTram maglev accelerator had muzzle velocities in excess of 8km/s, but the muzzle needed to be at least 4000m above sea level to get out of the thickest part of the atmosphere to reduce drag and heating. In any of these designs, the final product (a space launch vehicle) could have carried its own active cooling systems and ablative heat shield, increasing its survivability compared to your dumb projectile.
The Sprint missile only reached about 3.5km/s using an ablative heat shield (which reached temperature hot enough to melt tungsten) but was comparatively chunky as it has to carry much more stuff (multistage rocket, nuclear warhead, guidance equipment, etc) than a simple dumb projectile and it actively accelerated whilst in the thick atmosphere, so I suspect your bullet will be able to go a little faster than this.
Modern railguns can readily achieve muzzle velocities of 2.5km/s at very low altitude. The true answer therefore lies somewhere between about 3.5 and 9km/s, but its precise computation is left as an exercise to the reader.
There are additional issues, like initial heating being sufficient to cause melting, but drag then slowing the projectile below the critical velocity, and so on and so forth. Rocket science is hard, so you'll have to make do.