Can a mail carrier refuse to deliver mail because of a dog?

Mail carriers are not required to deliver mail if there is an "immediate threat" to the carrier, mail security, or postal property.

I don't know what a "pincher bug" is, exactly, so I can't say whether or not they would constitute an "immediate threat". However, if the carrier believes they're an immediate threat, they're within their rights to opt not to deliver your mail until the situation is corrected.