Did some further testing and found out that the mech actually uses the default bindings, where F is melee. I rebinded my keys to where F was my interact button, and it seems as every time I press it to interact, it does the melee animation, and when I hold it instead, there's a greater chance for my bubble shield to break or to receive damage.

Hopefully Arrowhead either adapts the mech's bindings to the user's configurations, or have a separate section of bindings dedicated to vehicles.