You reverse the time? If there is a random force term in your governing equations (e.g., Langevin dynamics), is the random number sequence also strictly reversed?
I doubt you really understand what you are talking about. Liapunov exponent measures how fast two trajectories deviate from an initial tiny difference. If your MD simulation is truly "microscopic bit-reversible", there is no initial difference at all. How does Liapunov exponent work here?
By the way, what kind of thermostat are you talking about? There are many different ways to implement a thermostat in MD simulations. You might be talking about Langevin dynamics, since you mentioned "noise source". But Langevin dynamics is not truly "microscopic", because some degree of freedoms are omitted. After all, that's why a random force term is introduced.