Band-structure calculations were performed using the spin-polarized relativistic Korringa–Kohn–Rostoker (SPR-KKR) band-structure method, determining intrinsic magnetic properties, such as magnetic moments, magnetocrystalline anisotropy energy (MAE), and Curie temperatures, of Fe5−x−yCoxMySiB2 (M = Re, W) alloys. The general gradient approximation (GGA) for the exchange–correlation potential and the atomic sphere approximation (ASA) were used in the calculations. Previous studies have shown that for Fe5SiB2, the easy magnetization direction is in-plane, but it turns axial for Co-doping in the range 1 < x ≤ 2.5 (y = 0). Furthermore, studies have shown that 5d-doping enhances the MAE by enabling the strong spin–orbit coupling of Fe–3d and M–5d states. The aim of the present theoretical calculations was to find the dependence of the anisotropy constant K1 for combined Co- and M-doping, building a two-dimensional (2D) map of K1 for 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1. Similar theoretical 2D maps for magnetization and Curie temperature vs. Co and M content (M = W and Re) were built, allowing for the selection of alloy compositions with enhanced values of uniaxial anisotropy, magnetization, and Curie temperature. The magnetic properties of the Fe4.1W0.9SiB2 alloy that meet the selection criteria for axial anisotropy K1 > 0.2 meV/f.u., Curie temperature Tc > 800 K determined by the mean-field approach, and magnetization µ0Ms > 1 T are discussed.