It is now commonly known that using optimal response-adaptive designs for data collection offers great potential in terms of optimizing expected outcomes, but poses multiple challenges for inferential goals. In many settings, such as phase-II or confirmatory clinical trials, a main barrier to their practical use is the lack of type-I error guarantees and/or power efficiency, especially in finite samples. This work addresses this gap. Specifically, focusing on a novel test statistic defined on the randomization probabilities of the (randomized) adaptive design, we derive its finite-sample and asymptotic guarantees. Further theoretical properties are evaluated under a Bayesian response-adaptive design, which is commonly used both in clinical trial applications and beyond (e.g., recommendation systems or mobile health). The frequentist error control advantages of the proposed approach–also able to preserve expected outcome optimalities–are illustrated in a real-world phase-II oncology trial and in simulation experiments.