An apparatus is provided for a turbine engine. This turbine engine apparatus includes a turbine engine component that includes a sidewall and a cooling aperture. The cooling aperture includes an inlet, an outlet, a meter section, a diffuser section and a transition section between and fluidly coupled with the meter section and the diffuser section. The cooling aperture extends through the sidewall from the inlet to the outlet. The meter section is at the inlet. The diffuser section is at the outlet. The transition section is configured to accommodate lateral misalignment between the meter section and the diffuser section.