In this Letter, the pump wavelength dependence of the amplified spontaneous emission (ASE) and the threshold of stimulated Brillouin scattering (SBS) in a typical single-frequency continuous wave Er3+:Yb3+-codoped fiber amplifier is investigated numerically. The Er3+:Yb3+ system is modeled as coupled two- and three-level systems, linked by a Förster resonance energy transfer process and described by the corresponding rate equations. The evolution of the pump and signal power along the fiber is modeled by differential equations, taking into account the steady-state population densities. Since the absorption at 976~nm can exceed the Yb3+-to-Er3+ energy transfer rate in high-power operation, unsaturated gain at around 1.0~μm can generate excessive ASE. Off-peak pumping with commercially available pump diodes at 915 or 940~nm spatially distributes the energy over a longer distance. For the studied amplifier configuration, energy transfer bottlenecking is prevented without the onset of SBS.