I don't see how such a thing could exist. Rationally equivalent classes should define the same element in an intersection theory. But, on $\mathbb{P}^1$, you can have two different divisors of the same degree, which are both $\mathrm{Gal}(\overline{k}/k)$ stable but have different Galois actions, and they will be rationally equivalent.

If you try to abandon the condition that rationally equivalent classes define the same element, you will run into trouble with self intersections. For example, consider a conic $C$ in $\mathbb{P}^2$. Its self intersection has degree $4$, representing that, for a generic perturbation $C'$ of $C$, the intersection $C' \cap C$ has four points. However, the Galois action on those $4$ points will depend on which perturbation you choose.

It is possible that your question is really the more general one of ``how do people study the sort of questions studied in intersection theory, while keeping track of Galois action"? There is tons of research on this front. Some papers I like are

J. Harris, *Galois groups of enumerative problems*, Duke Math. J. **46** (1979), 685–724

R. Vakil, *Schubert Induction*

A. Leykin and F. Sottile, *Galois groups of Schubert problems via homotopy computation*

However, as you will see, these papers do not use the kind of generalized intersection theory you are visualizing.