This paper studies inference for the realized Laplace transform (RLT) of volatility in a fixed-span setting using bootstrap methods. Specifically, since standard wild bootstrap procedures deliver inconsistent inference, we propose a local Gaussian (LG) bootstrap, establish its first-order asymptotic validity, and use Edgeworth expansions to show that the LG bootstrap inference achieves second-order asymptotic refinements. Moreover, we provide new Laplace transform-based estimators of the spot variance as well as the covariance, correlation, and beta between two semimartingales, and adapt our bootstrap procedure to the requisite scenario. We establish central limit theory for our estimators and first-order asymptotic validity of their associated bootstrap methods. Simulations demonstrate that the LG bootstrap outperforms existing feasible inference theory and wild bootstrap procedures in finite samples. Finally, we illustrate the use of the new methods by examining the coherence between stocks and bonds during the global financial crisis of 2008 as well as the COVID-19 pandemic stock sell-off during 2020, and by a forecasting exercise.