There won't be a difference in strength. This is related to a common question asked by those new to special relativity (and who have poor source material): If the mass of an object increases with its speed, could a planet, e.g., ever become a black-hole by traveling closer and closer to the speed of light? Well, as we know, this is mistaken reasoning that goes to the heart of relativity. The rest-mass of the object is the physically relevant quantity here (which is the whole of the energy-momentum of the object in its rest frame). The velocity of an object is only defined *relative* to an observer, and different observers in more and more Lorentz-boosted reference frame, so how could some say there is a black hole and others say it's just a planet, including the observers on the planet itself.
But we can get a general relativistic description.
Let's first restrict our attention to test particle observers (that is, ignoring their contribution to stress-energy). Then the stress-energy tensor of the source object, like a passing planet, is a frame-independent thing. Each observer will have their own idea of what the components of the tensor are (different mixes of energy and momentum flows), but the tensor itself is an object that is independent of these differing viewpoints. Therefore the spacetime curvature and geodesics on this spacetime are geometric objects (frame-independent) even though different observers will naturally *describe* the components of these quantities *in different ways*.
Now instead of being a test particle, suppose you're an observer on one planet passing another. Now your planet is contributing to the stress-energy distribution in spacetime, but with respect to the reference frames issue, nothing has changed. If you are on the planet, that is a particular observer frame and you measure the contribution due to your planet's mass-energy. That gives the components of your planet's stress-energy tensor in your frame. You then turn to the other planet, which is moving by at that moment, and you measure its apparent energy content and momentum through your frame. That gives the components of the stress-energy tensor of the second planet in your frame.
If that same planet came by at a higher relative speed, it's like being a test particle in a new Lorentz frame (a boosted frame). The physical results encoded in Einstein's equations should therefore be the same. Your stress-energy components haven't changed, while those of the passing planet *have changed* in the usual way for a boosted reference frame. The geometrical objects that are the stress-energy tensors of the two planets are the same objects after boosting, though that of the passing planet will look "distorted" from your point of view (that is, its components change). This is the same idea behind the 4-velocity. No matter what frame you go to the 4-velocity vectors of the two planets are what they are...it's just that they look relatively rotated from your particular reference frame of choice. Anyway, since the fundamental stress-energy tensors of the planets have not changed in this sense, the spacetime curvature and therefore gravitational strength will be the same, though you describe the two situations with different component values in your coordinate frame. In contrast, if you add more rest-mass to the other planet, the stress-energy tensor itself is changing (in any fixed Lorentz frame), and so the spacetime curvature would change.
Despite these arguments surrounding spacetime objects (4-velocity, stress-energy tensor, curvature tensor,...) being frame-independent, note that relatively boosted test particle reference frames obviously have different trajectories in space (just consider the various trajectories of objects near Earth's surface).
