The DDK system has gained increasing attention in recent research due to its potential to contain a three-hadron bound state. This paper utilizes an extension of the nonrelativistic effective field theory and the finite volume particle-dimer framework to derive Lorentz invariant quantization conditions for the DDK three-body system. Using current model input conditions, the finite volume energy spectrum of the DDK three-body system was calculated. This new calculation incorporates relativistic kinematics, allowing it to be applicable across a broader energy range starting from the threshold. In this work, we present a comprehensive O(p2) calculation. The spurious pole is effectively subtracted within the framework of relativistic kinematics. The spectra in the moving frame are also obtained. These analyses provide a broader testing ground for future lattice simulations. Based on this framework, it is expected to reveal more detailed properties of the DDK system and other three-hadron systems, if various higher-order effects, such as the isospin I=1 DK interactions and the coupled-channel effects are introduced. Published by the American Physical Society 2024