More precisly, the example I am interested most, is whether there are compactly supported smooth solutions of the equation $\Delta_g f=\delta\omega$ for some compactly supported smooth 1 form omega.

Thanks, I will check it out! Also, I just realized, my precise formulation also makes not a lot of sense since $\mathrm{ker}(L\vert_{c})$ is probably empty for many operators by unique continuation

Indeed, that was also what I was wondering. It seems to me that this product in this context has to be known for a much longer time. For example, Kühnel mentions in his book on differential geometry that the Kulkarni-Nomizu product was known as "double transvection" in Ricci calculus. So, this is how I got curious in how this product came to the name "Kulkarni-Nomizu" in the first place. The name seems to be already in use for quite some time. For example, Besse already used it in his book on Einstein manifolds from the 80s.