This is not what you asking, but maybe you want to know.

1. $\mathbb C\mathrm P^2$ does not admit a metric with splitting tensor in your sense. It has a non-zero Pontryagin number and any Pontryagin number can be expressed as an integral of some function of curvature tensor which is zero for splitting tensors. (In fact such a function is zero on any tensor which has orientation reversing symmetry and clearly splinting tensors are among them.)

2. Any riemannian metric on an open disc can be $C^0$ approximated by $C^\infty$ metrics with splitting curvature tensor. This follows from Nash--Kuiper embedding theorem.

This is not what you asking, but maybe you want to know.

1. $\mathbb C\mathrm P^2$ does not admit a metric with splitting tensor in your sense. It has a non-zero Pontryagin number and any Pontryagin number can be expressed as an integral of some function of curvature tensor which is zero for splitting tensors. (In fact such a function is zero on any tensor which has orientation reversing symmetry and clearly splinting tensors are among them.)

2. Any riemannian metric on $\mathbb S^n$ can be $C^0$ approximated by $C^\infty$ metrics with splitting curvature tensor. This follows from Nash--Kuiper embedding theorem and part (1) in the answer of Thomas Richard.