Testing if the mean of a Laplace distributed random variable is zero

Question

Suppose I have a sample $x=(x_1,x_2,...,x_n)^T$ of size $n$. Suppose I perdormed a statistical test and successfully showed that the sampling comes from a Laplace distribution with parameters that are estimated using the Maximum Likelihood estimates:
For a random variable $X \sim Laplace(\mu,\sigma)$ I define the PDF as $$f_X(x)=\frac{1}{\sigma\sqrt{2}}\exp{\left(-\frac{|x-\mu|}{\sigma}\sqrt{2}\right)}$$ And the Maximum Likelihood estimates are $$\hat{\mu}=x_{med}, \hat{\sigma}=\frac{\sqrt{2}}{n}\sum^n_{i=1}{|x_i-x_{med}|}$$ Where $x_{med}$ is the sample median.
However, the estimated parameter $\mu$ seems to be very small and I want to test a hypothesis $H_0:\mu=0$ against the alternative $H_1:\mu=\mu_1>0$ for an unknown std $\sigma$ using a likelihood ratio test. I compute the log of the ratio: $$\ln\left(\frac{L(x|H_1)}{L(x|H_0)}\right)=\frac{\sqrt{2}}{\sigma}\left(\sum_{i=1}^{n}{|x_i|}-\sum_{i=1}^{n}{|x_i-\mu_1|}\right)$$ But now I do not really know what to do next. How do I arrive at some statistic to test my hypothesis?
I know that if iid $x_i \sim Laplace(\mu,\sigma)$, then $\frac{2\sqrt{2}}{\sigma}\sum_{i=1}^{n}{|x_i-\mu|} \sim \chi^2_{2n}$, but I do not know how to exactly use this fact here. The second sum in the expression above seems to be getting in the way. Can anyone hint at what I need to do? Also, how do I factor in the fact that I do not know the value of $\sigma$ and only the ML estimate $\hat{\sigma}$ given above?
Edit: it seems to me that if we consider iid $x_i \sim Laplace(\mu,\sigma)$, then the statistic $\frac{2\sqrt{2}}{\sigma}\sum_{i=1}^{n}{|x_i-\mu|}$ can be used directly for hypothesis testing if the parameter $\sigma$ is known. However, I do not know $\sigma$ and need to find the distribution of $\hat{\sigma}$ in order to create a new statistic that would factor this uncertainty.