Until an expert in this field gives a rigorous answer to your question, I will give you my view.
If I understand correctly your questions, you are on the one hand asking about other types of existing exchange interactions between virtual particles and real particles, and on the other hand you claim that using the usual virtual photon picture for the momentum exchange between two billiard balls, would undermine Feynman's least action principle.
In all the Feynman diagrams I have seen so far, virtual photons are the element of choice and not phonons (as you ask in comments), as the idea is to use electromagnetic fields to describe repulsion/attraction forces, momentum exchange, charged particle interactions and so on. Furthermore, in case of a simple momentum exchange between two objects, one doesn't deal with interatomic modes of oscillation, so phonons do not really fit our case here anyway.
As for the least action principle, it does not actually matter how large the energy exchanged between the two objects is, because what the least principle action tells you, is that only paths (here for the virtual photons) close to the least action contribute to the final probability amplitude of paths, whose modulus squared here corresponds to the probability density of the virtual photons being absorbed in the shell x that would require the least action. This then means that for any amount of total energy in terms of virtual photons that has to absorbed for the momentum exchange to take place, the frequency of the constituent virtual photons is then defined based on the shell-energy that has contributed most to the overall paths' amplitude (which will also dictate the number of virtual photons needed).
So following this logic, the further off shell interactions are most likely to contribute least because of the least action principle, without the need of another type of exchange interaction.
It may help to make contrast between this scenario and that of the electron-nucleon interaction, where long wavelength virtual photons will only see the whole nucleon and shorter ones will be able to see and interact with the quarks inside.
Anna v's answer in an older post also contains relavant bits of explanations that you may want to check out. One other source would be this.