We develop a combined Lagrangian-Eulerian method for transient fluid-structure interaction problems. Based on the ghost fluid framework for improving interface tracking accuracy between a fluid (hot rocket exhaust plume) and a high strain rate deforming solid (rear cover of a vertical launch system), the numerical coupling between the two media ensures an accurate description of the flexible structure. A nine-node quadrilateral element based on total Lagrangian formulation is used, while the hydrodynamic finite difference method is used for the supersonic exhaust plume that forms a complex flow within the plenum. The Lagrangian, Eulerian, and fluid-structure interaction coupling methods are verified by ANSYS results and related theories. A two-dimensional simulation of the full vertical launch system operation mode is conducted. This requires an accurate reproduction of the complex flowfield generated by the rapid rear cover opening under a high-pressure plume during rocket launch. This fluid-structure interaction problem solution may be used for future design upgrades when a vertical launch system is exposed to unusually harsh interactive gas and structure conditions.