Well first, as a simple matter of fact, the momentum density (and the rest of the stress-energy tensor) *does* vary depending on position. This is because the universe is not filled up with a homogeneous fluid moving everywhere with the same speed in the same direction. Look around you to see that this is not the case.

But more importantly, this comment of yours is exactly my point about assumptions. Where, exactly, is it that you are getting your ideas from about what the gravitational field should or shouldn't depend on? Where are you getting ideas about how sensitive it should or should not be to local variability of momentum density, in any situation, or anything else for that matter? In GR, the *only* source of knowledge we have about that sort of thing is from the Einstein field equation and the (often approximate) solutions people have been able to derive for it in specific cases over the years. That includes solutions showing that it can make a difference if you have a glob of matter that is rotating v.s. not rotating. This is not in conflict with anything because there simply does not exist some alternative method of reasoning you can follow to derive that the gravitational field should behave any differently. It is just you who seem to be approaching GR with a lot of beliefs about what sort of behaviour the Einstein field equation should predict without having mathematically studied or solved it.

I'd already commented on the complexity of the Einstein field equation a couple of posts ago, but just to emphasise one thing about it here: like I said before, it is a *nonlinear* differential equation. In more physical terms, that means it does not obey the superposition principle. That means that you absolutely cannot understand the gravitational field produced by a complicated system by reducing it to its parts. GR is not like Newtonian gravity where you can find the gravitational field produced by any mass distribution by adding the gravitational fields produced by all its constituent parts. For example, you can't derive the gravitational field around two nearby masses in space just by adding two Schwarzschild metrics, or anything so simple. You would basically need to solve the Einstein field equation for that specific situation, starting from scratch, if you wanted to do that.