The underlying chemical and physical mechanism of avian navigation is an important issue of broad interest. One of the most famous candidates is the radical-pair mechanism, which shows that the magnetoreception is achieved by detecting the amount of chemical-reaction product from the singlet state. In the hypothesis, the surrounding nuclear spins play an important role as inducing the coherent transition between the singlet and triplet states. Recently, it was suggested that a multiradical model beyond two radicals can also realize magnetoreception without the assistance of nuclear spins. Inspired by this discovery, we explore the amount of the singlet recombination product in a multiradical model, with a radical bath described by the Lipkin-Meshkov-Glick (LMG) model, which was originally proposed for quantum phase transition (QPT). We show that the sensitivity of the bionic compass can be improved at the critical point. Our results may pave the way for the exploration of the design principle of the bionic compass.