  Using high-throughput automation of ab-initio impurity embedding simulations,
we created a database of $3d$ and $4d$ transition metal defects embedded into
the prototypical topological insulator (TI) Bi$_2$Te$_3$. We simulate both
single impurities as well as impurity dimers at different impurity-impurity
distances inside the TI. We extract changes to magnetic moments, analyze the
polarizability of non-magnetic impurity atoms via nearby magnetic impurity
atoms and calculate the exchange coupling constants for a Heisenberg
Hamiltonian. We uncover chemical trends in the exchange coupling constants and
discuss the impurities' potential with respect to magnetic order in the fields
of quantum anomalous Hall insulators and topological quantum computing. In
particular, we predict that co-doping of different magnetic dopants is a viable
strategy to engineer the magnetic ground state in magnetic TIs.
