This paper deals with a novel formulation of a nonlinear reduced-order computational model for analyzing the non-linear vibrations of a linear viscoelastic structure with weak nonlinear geometrical effects, coupled with a linear acoustic liquid with sloshing and capillarity on the free surface. The model proposed is derived from the one used in fluid-structure interaction for linear systems, for which the analysis of the acoustic-sloshing-capillarity phenomena is efficient thanks to the use of a projection on the linear modes of the linear acoustic liquid and on the sloshing modes with capillarity. Concerning the construction of the vector basis for the structure, it is proposed to use a POD approach for the viscoelastic structure with weak nonlinear geometrical effects and taking into account the added mass induced by the liquid. The methodology for constructing the vector bases of the admissible sets and for obtaining the non-linear ROM are detailed. The computational nonlinear ROM that is presented can directly be used for analyzing the vibrations of such fluid-structure systems using commercial finite element softwares for computing the vector bases and the projections.