Let’s state an underlying assumption of your question:
The excitations of the force carrying fields are quantised and can be thought of as particles.
This is one of the basic assumptions of particle physics and is the key to understanding and predicting the behaviour connected with electromagnetism and the weak and strong nuclear forces.
The quantum field theory approach to gravity is not nearly as successful. Arguably it just doesn’t work. One of the crucial difficulties stems from the following fact. Quantum field theories are defined on a background geometry, but the gravitational field is the geometry.
A partial work around is to treat the problem pertubatively: assume a background geometry, and treat the small variations around it as a separate field. This perturbative approach is already hard in the classical field theory of gravity (General Relativity) and is non-renormalisable in the quantum version (someone correct me if I’m wrong).
But even if it did work, we still wouldn’t be able to answer your question i think. One would still have to assume the background geometry to be that of our universe (expanding, isotropic, homogeneous) and thus the photon redshift would still be an effect of geometry, not of particle interactions.
In conclusion, to my knowledge, there is no way to understand gravity in the form of gravitons in a way that is self-consistent and would be able to answer questions like yours.