I have used CLR on both metal bergs and Links without any issues.
CLR is a Calcium Lime Rust cleaner/de-scaler often used to de-scale coffee machines and the like.
Most decent coffee machines use both copper and stainless parts and various rubber and cork parts, so little danger of damaging a mouthpiece with it.
Clean thoroughly afterwards with soap and water.
Denture cleaner is also a decent cleaner for mouthpieces.
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