Data-efficiency is crucial for autonomous robots to adapt to new tasks and environments. In this work, we focus on robotics problems with a budget of only 10-20 trials. This is a very challenging setting even for data-efficient approaches like Bayesian optimization (BO), especially when optimizing higher-dimensional controllers. Previous work extracted expert-designed low-dimensional features from simulation trajectories to construct informed kernels and run ultra sample-efficient BO on hardware. We remove the need for expert-designed features by proposing a model and architecture for a sequential variational autoencoder that embeds the space of simulated trajectories into a lower-dimensional space of latent paths in an unsupervised way. We further compress the search space for BO by reducing exploration in parts of the state space that are undesirable, without requiring explicit constraints on controller parameters. We validate our approach with hardware experiments on a Daisy hexapod robot and an ABB Yumi manipulator. We also present simulation experiments with further comparisons to several baselines on Daisy and two manipulators. Our experiments indicate the proposed trajectory-based kernel with dynamic compression can offer ultra data-efficient optimization.