A few comments about pre-requisites to teaching students to speak Latin.
Concerning the difficulty to reconstitute the accents of Ancient Latin, the question of "what should be the correct accent" should not a subject of concern (English teachers do not worry that accents of the 16th century were markedly different from those of today, even when their students are being requested to read Shakespeare).
In the end, the accent used to speak Latin should not be important, as long as it remains understandable to another speaker (e.g. as when a foreign langue students choses between American or British accents). We could consider that we have a "received pronunciation" and leave it at that.
By contrast, the various accents used to speak Latin in recent times might even constitute a fun subject for students. For style used by the Finnish Radio, here are world news in Latin, "Nuntii Latini" (lots of fun to listen to). Besides, that could give subjects of conversation for students!
Use as Active Language
Latin has remained a very active "educated" language (used in a context of diglossia) actively used throughout the Middle Ages up to recent times, both in written texts and speech. It would be a pity not to introduce students to the immense corpus of very interesting material (scientific, philosophical, religious) written in Latin during the Middle Ages and Modern Times. Since it was a common language, it gives students access to material from a vast number of countries which would have been inaccessible to them if they had been written in the vulgar language of that country. A well known text is the ascension of Mount Ventoux in France by Francesco Petrarca (1336).
Most importantly, the fact that beside the lofty poems and philosophical texts by "litterary stars" such as Cicero or Virgil there are many mundane sources (letters, court minutes, etc.), could "de-sacralize" the language and show that it can be used correctly even by less expert persons. For further study, there should be many dialogues (whether literary or transcriptions).
Arguably, this would not teach to speak, but at least, it would show how people used the language in mundane situations.
Omnia dici possunt Latine. A key part of teaching any spoken language would be to provide glossaries for daily situations (it would be hard to have a conversation today if we couldn't name a car, a computer, etc.). Hence students should be taught how to call them, as well as names of places. A student of Angelopolis might thrilled to learn that his region has been so instrumental in popularizing pelilculae cinematographicae and computatra...