To complete on @srtee remarks, the units on the one hand and the effect of the brownian damping constant on the other are two different things.
Questions regarding comprehension of lj units are common on the forum (see this thread for example), you can look up in the archive for more detailed discussions and physics textbooks/papers for common usages.
In the units command manual page \tau is the time unit notation in reduced units. It is not directly related to the (1 dimensional) diffusion time scale you’re defining using the same symbol which is a quantity you will get from your simulations. The former is just used as a unit for the latter.
However, as far as I understand what you try to achieve, nothing stops you from computing the said diffusion time scale (let’s call it \xi) and to express your simulation time in reduced \xi units \tau'=\frac{\tau}{\xi} in a post-process from your results. This is unrelated to the units you’re using. Again this is standard procedure you’ll find in the relevant physics literature or explained by more skilled colleagues.