We study the hard edge limit of a multilevel extension of the Laguerre $β$-ensemble at zero temperature. In particular, we show that asymptotically the ensemble is given by Gaussians with covariance matrix expressible in terms of the Fourier-Bessel series. These Gaussians also have an explicit representation as the partition functions of additive polymers arising from a random walk on roots of the Bessel functions. Our approach builds off of the one introduced in arxiv:2009.02006 and is rooted in using the theory of dual and associated polynomials to diagonalize transition matrices relating levels of the ensemble.