I found a mention of him, while he was still alive, in an Italian literary journal from 1710:
As you can see, here it is explained that the title of his work, מלאכת מחשבת, is also an acronym for משה חפץ שוכן בן-עיר טריסטי.
In addition, I found an earlier assertion that he was 100 years old when he published this work (the other post mentioned an Italian book from 1819). In Giovanni Bernardo de Rossi's great
"morì nel 1713 in età di 103 anni," which means that he passed away in 1713 at 103 years of age. de Rossi did not explain his source for this piece of information, but it evidently is the title page. What makes this even more amazing is that the very entry preceding the one for our subject is a short entry for his son who died in 1700 aged 17. If his father did die thirteen years later at 103, that would mean that his son was born to him when he was 73 years old. Not impossible, but not likely!
* While sort of true, I jest. The work cites שם הגדולים (pt ii) and ועד לחכמים consistently. But the fact remains that a great deal of the information in the book is just the שם הגדולים put into Italian (although it uses other sources, and de Rossi's own research). Incidentally, this same work mentions the Chida as being aged and still living in Livorno (this from the entry on his great-grandfather).