Reading a lot of Roman History, if the ice was there, as it is today, the movement of Roman merchants and Roman military would have been greatly curtailed. Armies were moved back and forth with a speed that would indicate that the ice as we know it today was not there.
Hannibal’s invasion was in 218 B.C., that was during the Roman Republic, not the Empire. Polybius and Livy’s accounts both state that Hannibal encountered ice. Must have fluctuated again by the time of the Roman Empire.
But even so he was no luckier; progress was impossible, for though there was good foothold in the quite shallow layer of soft fresh snow which had covered the old snow underneath, nevertheless as soon as it had been trampled and dispersed by the feet of all those men and animals, there was left to tread upon only the bare ice and liquid slush of melting snow underneath. The result was a horrible struggle, the ice affording no foothold in any case, and least of all on a steep slope; when a man tried by hands or knees to get on his feet again, even those useless supports slipped from under him and let him down; there were no stumps or roots anywhere to afford a purchase to either foot or hand; in short, there was nothing for it but to roll and slither on the smooth ice and melting snow.
The definition of a glacier is snow left over from the previous year. This account describes what it is like to try to climb a glacier without crampons.