The VHF Data Broadcast (VDB) data link is responsible for transmitting Ground Based Augmentation System (GBAS) corrections from the GBAS ground station to the aircraft. Thus, it is a major bottleneck for the evolution and security of GBAS. It provides limited bandwidth, range, only line-of-sight capabilities, and no cyber-security protections for the transmitted data. Overcoming these constraints is required for the future and calls for an alternative data link for GBAS. A promising candidate is the L-band Digital Aeronautical Communications System (LDACS). First demonstrations of secure GBAS over LDACS used the Timed Efficient Stream Loss-tolerant Authentication (TESLA) for broadcast authentication of GBAS data. In flight trials, the concept and support of TESLA secured GBAS via LDACS for GBAS services, supporting category II/III precision approach capabilities, were proven. In this paper, different ways are investigated to further optimize latency and security data overhead for an optimized transmission of TESLA secured GBAS packets via LDACS. Initial evaluations show how promising the different options are, especially in respect of a reduced latency of 55.45~ms compared to previous 632.98~ms. Further, it is shown how the developed concept for secure GBAS can also be applied to secure general broadcast applications over LDACS.