This paper studies the use of a multi-attribute auction in a communication system to bring about efficient relaying in a non-cooperative setting. We consider a system where a source seeks to offload data to an access point (AP) while balancing both the timeliness and energy-efficiency of the transmission. A deep fade in the communication channel (due to, e.g., a line-of-sight blockage) makes direct communication costly, and the source may alternatively rely on non-cooperative UEs to act as relays. We propose a multi-attribute auction to select a UE and to determine the duration and power of the transmission, with payments to the UE taking the form of energy sent via wireless power transfer (WPT). The quality of the channel from a UE to the AP constitutes private information, and bids consist of a transmission time and transmission power. We show that under a second-preferred-offer auction, truthful bidding by all candidate UEs forms a Nash Equilibrium. However, this auction is not incentive compatible, and we present a modified auction in which truthful bidding is in fact a dominant strategy. Extensive numerical experimentation illustrates the efficacy of our approach, which we compare to a cooperative baseline. We demonstrate that with as few as two candidates, our improved mechanism leads to as much as a 76% reduction in energy consumption, and that with as few as three candidates, the transmission time decreases by as much as 55%. Further, we see that as the number of candidates increases, the performance of our mechanism approaches that of the cooperative baseline. Overall, our findings highlight the potential of multi-attribute auctions to enhance the efficiency of data transfer in non-cooperative settings.