It isn't a surprise that José de la Cruz had a nickname. Names as common as his almost required a nickname, to help people distinguish WHICH of the many José de la Cruzes you were talking about.
What's surprising is that this José de la Cruz had a Mexican nickname.
In the 1832 document, this person's identity is, "José de la Cruz, alias 'el Guachinango.'"
Having an ear for regionalisms, I suspected it was Mexican. And sure enough, there is a town in the State of Jalisco in Mexico called Guachinango.
To make matters more interesting (or more complicated), guachinango can also mean a kind of fish (red snapper). In Cuban and Puerto Rican dialect, it can mean a clever person, a joker or a flatterer.
So why does this Chamorro guy have a nickname like this?
Well let's not assume he was Chamorro. In 1832, there very well could have been a Mexican, Cuban, Puerto Rican or just about any Latin American person living on Guam. But even if he were Chamorro, Mexican influence made a mark on the Chamorro language and culture in the 1700s. The Acapulco Galleons were passing through Guam until the time during Mexico's war for independence in 1815, not too long from 1832, the date of this document.
In some parts of the Philippines, gwatsinanggo means "shrewd" or "cunning," among other things, which follows one of the meanings of the word for in the Caribbean. The fact that the word made it even as far west as the Philippines makes it more credible that the word spread to the Marianas.
The fact is that our islands were always getting visitors from east and west and being influenced by them. Why José was called "El Guachinango" will remain a mystery.