Battering rams are designed for immobile targets, which are fixed against a structure. While difficult to move, the Spartan phalanx was not immobile.

If you ran a battering ram at the Spartan line, they would uniformly trot backwards until the battering ram had pushed its way into the Spartan line, and been surrounded and massacred, or the entire Spartan line would trot backwards, envelop the ram, and massacre the men.

The reason the Spartans couldn't be dislodged from the pass was because the force that Xerxes could apply was simply not enough to break through. Thousands of men were slaughtered because only a hundred or so men could face a hundred or so Spartans at a time.

Look at it analytically. If an enemy has a fantastic defensive position, and their entire objective is to simply hold that defensive position, then you can either go around them (which the Persians eventually did), or bring overwhelming firepower from a position which cannot be assaulted without leaving the defensive position.

If siege engines such as trebuchet or catapults were used, they could simply bombard the Spartan line with huge rocks and flaming logs until there is so much debris the phalanx cannot effectively defend a push, because before the Roman maniple, the phalanx was useless on broken terrain.

There are a bunch of ways the Spartans could be dislodged, but there's also a bunch of ways a skilled commander could have avoided conflict, and still won the day. Julius Caesar or Alexander would have had endless fun at Thermopylae, but Xerxes was a shit commander overconfident in his numbers.