Could the Mass and the Office be said in the classical pronunciation?
Yes. It would be a little strange, but, yes.
The way we speak Latin in the Church right now is based on how Italian is pronounced, or, how Romans pronounce Latin. That’s reasonable, given that we are talking about the language of the Roman Church. Who better than they?
Latin is pronounced in different ways, according to one’s background and nation. The English school system had a truly weird system. Germans do odd things with vowels. The French… well….
What we call now the Classical Pronunciation is more or less the fruit of research into how Latin might have been pronounced in the late Republic and early Empire, in the Gold and Silver Age of Latin literature. We extrapolate how things were pronounced by examining misspellings in inscriptions and other writings, along with morphology, etc.
That said, pronunciation was not uniform. North Africa was different from Italian peninsula. Just as is the case today, pronunciation surely varied within cities.
Also, consider that between the Gold and Silver Ages of literature and, say, the time of Augustine of Hippo, there is not only a gulf of distance but a gulf of centuries. Early Modern English in, say, the plays of Shakespeare, sounds a bit foreign to our ears until we adjust. Of course, reconstructed pronunciation helps us to hear rhymes and puns in Shakespeare. Reconstructed Classical Latin pronunciation also helps in the learning of Latin. Think about the principle parts of ago and how they are pronounced in Ecclesiastical and Classical systems.
People who are interested in solid scholarship on reconstructed pronunciation of classical Latin, and Greek for that matter, can look into the standard work of W. Sydney Allen, Vox Latina. Also see E.H. Sturtevant. They both also wrote on Greek.
I did a lot of this sort of thing in grad school, but it has been while. I am sure there are now some new resources.