Deep Learning (DL) has become essential in various robotics applications due to excelling at processing raw sensory data to extract task specific information from semantic objects. For example, vision-based object-relative navigation relies on a DL-based 6D object pose predictor to provide the relative pose between the object and the robot as measurements to the robot's state estimator. Accurately knowing the uncertainty inherent in such Deep Neural Network (DNN) based measurements is essential for probabilistic state estimators subsequently guiding the robot's tasks. Thus, in this letter, we show that we can extend any existing DL-based object-relative pose predictor for aleatoric uncertainty inference simply by including two multi-layer perceptrons detached from the translational and rotational part of the DL predictor. This allows for efficient training while freezing the existing pre-trained predictor. We then use the inferred 6D pose and its uncertainty as a measurement and corresponding noise covariance matrix in an extended Kalman filter (EKF). Our approach induces minimal computational overhead such that the state estimator can be deployed on edge devices while benefiting from the dynamically inferred measurement uncertainty. This increases the performance of the object-relative state estimation task compared to a fix-covariance approach. We conduct evaluations on synthetic data and real-world data to underline the benefits of aleatoric uncertainty inference for the object-relative state estimation task.