Recently, we proposed a way to engineer a flexible acoustomechanical coupling between the center-of-mass motion of an isolated micromagnet and one of its internal acoustic phonons by using a magnon as a passive mediator [C. Gonzalez-Ballestero, J. Gieseler, and O. Romero-Isart, Phys. Rev. Lett. 124, 093602 (2020)]. In our approach, the coupling is enabled by the strong magnetoelastic interaction between magnons and acoustic phonons which originates from the small particle size. Here, we substantially extend our previous work. First, we provide the full theory of the quantum acoustomagnonic interaction in small micromagnets and analytically calculate the magnon-phonon coupling rates. Second, we fully derive the acoustomechanical Hamiltonian presented in Gonzalez-Ballestero et al. Finally, we extend our previous results for the fundamental acoustic mode to higher-order modes. Specifically, we show the cooling of the center-of-mass motion with a range of internal acoustic modes. Additionally, we derive the power spectral densities of the center-of-mass motion which allow us to probe the same acoustic modes.