Due to the interaction between the topological defects of an order parameter and underlying fermions, the defects can possess induced fermion numbers, leading to several exotic phenomena of fundamental importance to both condensed matter and high-energy physics. One of the intriguing outcomes of induced fermion numbers is the presence of fluctuating competing orders inside the core of a topological defect. In this regard, the interaction between fermions and skyrmion excitations of an antiferromagnetic phase can have important consequences for understanding the global phase diagrams of many condensed matter systems where antiferromagnetism and several singlet orders compete. We critically investigate the relation between fluctuating competing orders and skyrmion excitations of the antiferromagnetic insulating phase of a half-filled Kondo-Heisenberg model on a honeycomb lattice. By combining analytical and numerical methods, we obtain the exact eigenstates of underlying Dirac fermions in the presence of a single skyrmion configuration, which are used for computing the induced chiral charge. Additionally, by employing this nonperturbative eigenbasis, we calculate the susceptibilities of different translational symmetry breaking charges, bond and current density wave orders, and translational symmetry preserving Kondo singlet formations. Based on the computed susceptibilities, we establish spin Peierls and Kondo singlets as dominant competing orders of antiferromagnetism. We show favorable agreement between our findings and field theoretic predictions based on the perturbative gradient expansion scheme, which crucially relies on the adiabatic principle and plane-wave eigenstates for Dirac fermions. The methodology developed here can be applied to many other correlated systems supporting competition between spin-triplet and spin-singlet orders in both lower and higher spatial dimensions.