Very Massive Relativistic Body

The black hole solution you are referring to is the Schwarzschild solution that applies to a static centrally symmetric object. If the object is moving in your reference frame, it is not static, so this solution does not apply in your coordinates. In other words, the Schwarzschild spacetime is not Lorentz invariant just like most anything in General Relativity.

The easiest way to resolve this is to describe the object in its rest frame and then use the equivalence principle that physics does not depend on the reference frame. If the object is not a black hole in its own rest frame, then, according to the equivalence principle, this object is not a black hole in any reference frame.