The computational and geometrical complexity of flow dynamics yields to the need of powerful yet simple numerical tool. Amongst different existing methods, the lattice Boltzmann method (LBM) has recently acquired a growing interest due to its low computational cost, relatively simple parallelisation and an ability to handle complex geometries. Particularly, these advantages arise thanks to a low-order spatial stencil which is used in lattices with few velocities. However, this leads to an inability of classical LBM to deal with thermal and compressible flows. The coupling between LBM and a finite difference discretisation of entropy equation has been a successful attempt to tackle this limitation. This has been further extended in proposed fully conservative hybrid model to solve the energy equation separately from mass and momentum. While this development indeed led to the improvement of the solution in terms of the correct jumps relations recovery across the shock waves, the stabilisation of the method for the applications where flow discontinuities are observed remains a challenge. This work aims to set an appropriate framework to apply LBM to fully compressible flows with discontinuities due to the strong shock waves. This is twofold. Firstly, we establish a new class of LBM basic discretisation based on the explicit kinetic schemes of arbitrary high order. Secondly, we implement a non-linear MOOD stabilisation technique aiming to restore the stability in the critical areas of the solution while not violating the accuracy of the smooth regions. The stability limits of proposed method will be presented along with precision studies based on the classical validation problems such as thermal Couette flow, entropy spot convection and Sod shock tube.