It is possible that the universe has naturally a slight curve of spacetime, even in a place where there is not a massive object, and that this curve is responsible for the perception of the relative velocity between two bodies? For example: The body A and body B, has as its starting point the C point and arrival point D. The speed of A relative to C is 200 m / s and B to C, 100 m / s. It may be that the body A is in a less curved position of spacetime than that B and therefore the time interval and the spatial displacement experienced by A to arrive at D is less than that experienced by the B also to get D? I think of the possibility that at least 4-dimensional (3 space and 1 time) are curved, so that the universe is somewhat like a multidimensional bubble. From this point of view, then the body A would be traveling for the inner path (inside the bubble, so to speak) than the body B. The path from A to reach D, is shorter in time and space in the direction of its movement, than the path from B to get to D. That would not explain the space contraction and time dilation? I believe that only one spatial dimension is involved in the event, that is, the dimension in which the body moves. Therefore in the remaining two spatial dimensions, bodies with different velocities could be extremely near.
Velocity is not simply related to spacetime curvature. If it was you would struggle to explain how two objects can be extremely close, and therefore presumably experiencing the same curvature, but still have different velocities.
At the risk of over simplifying, spacetime curvature is normally related to a relative acceleration rather than a relative velocity.
However, it is indeed possible that the universe has a small spacetime curvature. Our universe is (approximately) described by the FLRW metric, and in this metric the curvature is an adjustable parameter. Experiment shows the universe is flat to about 1%, and it's certainly possible that the curvature is non-zero but is too small for us to measure.