Let me have a bucketful of ice brought.

Literal

Ice [object-を] bucket one-cup bringing [causative-させてくれ].

持ってこさせてくれ layers causative + benefactive: させる ('make someone do') + てくれ ('do for me'). The speaker is asking permission to have someone else bring ice. バケツ1杯 means 'one bucketful.'