This work addresses a reactive jamming attack on the low-latency messages of a victim, wherein the jammer deploys countermeasure detection mechanisms to change its strategy. We highlight that the existing schemes against reactive jammers use relays with instantaneous full-duplex (FD) radios to evade the attack. However, due to the limitation of the radio architecture of the FD helper, instantaneous forwarding may not be possible in practice, thereby leading to increased decoding complexity at the destination and a high detection probability at the adversary. Pointing at this drawback, we propose a delay-aware cooperative framework wherein the victim seeks assistance from a delay-aware FD helper to forward its messages to the destination within the latency constraints. In particular, we first model the processing delay at the helper based on its hardware architecture, and then propose two low-complexity mitigation schemes, wherein the victim and the helper share their uplink frequencies using appropriate energy-splitting factors. For both the schemes, we solve the optimization problems of computing the near-optimal energy-splitting factors that minimize the joint error rates at the destination. Finally, through analytical and simulation results, we show that the proposed schemes facilitate the victim in evading the jamming attack whilst deceiving the reactive adversary.