Regardless of whether you consider yourself as moving, or the object as moving, the measurements you can actually make will turn out the same.
The "gravitational field" of a hyper-relativistic object will approach that of an impulsive gravitational plane wave, similar to the way the electric field of a hyper-relativistic charge approaches an impulsive electromagnetic plane wave.
"Gravitational field" is an ambiguous term, in this case I am using it to mean tidal gravity, something you can actually measure (i.e. for example with a Forward mass detector, or a gravitational gradient meter (typically a rotating cruciform gravitational gradiometer) like those they use to survey for oil with.
http://www.bellgeo.com/tech/technology_theory_of_FTG.html
This is probably NOT what you're used to thinking of as the gravitational field, but if you consider the problem of the Moon's gravitational field effect on the Earth, you can perhaps see that we don't measure the Newtonian field directly, what we actually observe are time-varying tidal forces.
