However, whether that translates to a loss in momentum is a bit more fuzzy and I really can't tell whether it does or doesn't. Although I'm far from being a physicist, so hopefully someone more knowledgeable chimes in to enlighten us...
But I'm curious: what is your reasoning for asking whether photons have a mechanism to lose momentum as a consequence of them affecting the curvature of spacetime? It's not at all obvious to me the relationship between these two concepts.
Yes, redshifted photons lose momentum. Momentum (really the stress-energy tensor) is conserved locally, and along trajectories that preserve the metric, but global momentum conservation in GR isn't even well-defined.
This is not true, elastic scattering is very common.