We obtain a CLT for $\log|\det(M_n-s_n)|$ where $M_n$ is a scaled Laguerre $β$ ensemble and $s_n=d_++σ_n n^{-2/3}$ with $d_+$ denoting the upper edge of the limiting spectrum of $M_n$ and $σ_n$ a slowly growing function ($\log\log^2 n\llσ_n\ll\log^2 n$). In the special cases of LUE and LOE, we prove that the CLT also holds for $σ_n$ of constant order. A similar result was proved for Wigner matrices by Johnstone, Klochkov, Onatski, and Pavlyshyn. Obtaining this type of CLT of Laguerre matrices is of interest for statistical testing of critically spiked sample covariance matrices as well as free energy of bipartite spherical spin glasses at critical temperature.