Due to the long propagation delay inherent in underwater acoustic transmissions, forward error correction (FEC) is a preferable technique for end-to-end reliability control in relay-aided underwater acoustic communications (RA-UAC). Among all error-correcting codes, rateless fountain codes, such as Luby Transform (LT) codes, are attractive for their low computational complexity and rate adaptability to channel fading dynamics. For dual-hop communications, decomposed LT (DLT) codes have also been developed in the literature. However, when applying DLT codes in RA-UAC, one also needs to ensure small end-to-end communication latency and flexible computation cost balance between the source and the relays, in order to cope with the long propagation delay and heterogeneous node energy. For these purposes, we propose hybrid DLT (h-DLT) codes in this paper. In h-DLT codes, data encoding is performed in hybrid modes: cooperative DLT mode and direct LT mode. By choosing different combination ratio, the coding system can flexibly assign different computation load to the two DLT encoders. Based on the analysis of the resultant degree distribution of h-DLT codes, we develop algorithms to determine the encoding degree distributions of each mode of an h-DLT code with a specific computation requirement. Additionally, an h-DLT based RA-UAC (hR-UAC) protocol is proposed. To illustrate the system performance, we will evaluate the decoding probability and computation cost of h-DLT codes by comparing with primitive DLT codes and LT codes. In addition, analysis will be shown to demonstrate the benefits of the hR-UAC system in terms of both the relay computation cost and the end-to-end transmission latency.