Trumpet

This was a random gift and did not come with mouthpiece.

The one in the picture is a Fluegel mouthpiece, which isn't a good fit.

Would a piccolo mouthpiece be suitable?

I suppose just go with what you usually use, don't see a reason to change. If you've got a mouthpiece that work better with high resistance stuff like mutes then that would be a good option if you really want to switch.