Relaxed semantics have been introduced to increase the achievable parallelism of concurrent data structures in exchange for weakening their ordering semantics. In this paper, we revisit the balanced allocations $d$-choice load balancing scheme in the context of relaxed FIFO queues. Our novel load balancing approach distributes operations evenly across $n$ sub-queues based on operation counts, achieving low relaxation errors independent on the queues size, as opposed to similar earlier designs. We prove its relaxation errors to be of $\mathcal{O}(\frac{n \log\log n}{\log d})$ with high probability for a collection of possible executions. Furthermore, our scheme, contrary to previous ones, manages to interface and integrate the most performant linearizable queue designs from the literature as components. Our resulting relaxed FIFO queue is experimentally shown to outperform the previously best design using balanced allocations by more than four times in throughput, while simultaneously incurring less than a thousandth of its relaxation errors. In a concurrent breadth-first-search benchmark, our queue consistently outperforms both relaxed and strict state-of-the-art FIFO queues.