The LHCb detector relies on a fully software-based trigger system to process proton-proton collisions at 30 MHz. This has been possible due to the development of Allen, a GPU implementation of the first stage of the High Level Trigger deployed on approximately 500 GPU cards which reduces the input data rate from about 4 TB/s to 150 GB/s by exploiting parallelism in intensive algorithms such as particle tracking and primary-vertex reconstruction. Allen has continuously evolved to improve the efficiency of its algorithms and incorporate novel reconstruction techniques in preparations for future upgrades. One such new development is the reconstruction of high-multiplicity secondary vertices, enhancing the selection of particles decaying into many final state products. These signatures are relevant for detector material mapping and for precision measurements with complex decay topologies, either in Standard Model analyses or in searches for New Physics. This work presents the design and implementation of a GPU-optimised multi-track vertex reconstruction algorithm capable of operating at the full 30 MHz collision rate during Run III. Its performance is evaluated using simulation, with particular emphasis on reconstruction efficiency and throughput impact. We demonstrate the feasibility of performing multi-body vertex reconstruction at the first trigger level in LHCb and establish the foundations for a robust and high-throughput implementation in preparation for Run IV data-taking conditions.