I think this came out pretty tidy:
If time has no beginning, then infinite time has passed.
If entropy increases monotonically with time (2nd law of thermo), then after infinite time it must either have diverged to infinity or converged to a maximum value.
Case 1: The heat death of the universe has not occurred, therefore entropy is not infinite.
Case 2: Internal combustion engines work, therefore entropy has not reached a maximum value.
Therefore time had a beginning. (modus tollens)
This still allows for appeals to very large or infinite volumes of space for life to arise, which then has to contend with the Fermi paradox.