We don't know. What's mentioned is informative, but certainly not decisive. Language is way too noisy for simple conclusions. E.g., spelling errors aren't exclusively based on phonetic similarity, and even if they were, their absence proves little.
Latin followed the alphabetic principle with few exceptions (U and J were later invented to fix the main exceptions). We know almost exactly how it was pronunced because of that and remarks contemporaries recorded on how speech sounded. For example, R was called the littera canina because of the trill. It is especially noticeable when you have two of them together like in terra.
The main thing we do not know is how regnum was pronounced. We know it was either of two options for interpreting gn and a third choice is that both were acceptable. People are also unsure how 4 of the short vowels sounded. Some say that they have slightly different sounds while others say that they are just short versions of the long vowels like the other two (A and Y). It is possible both variations co-existed.
We also know that western Romans often mispronounced Y as I since they had trouble rounding their lips for Y. Y had been introduced for transliteration of Greek loanwords, so it was not a native sound for the western Romans.
There is more to a language than knowing how to pronounce each individual letter. You can easily see it if you take any modern language, write a text in it down in IPA and ask a linguist unfamiliar with the language itself read it. It will still sound very alien to a native speaker. And this is with a live language we have full knowledge of.
The result of this exercise depends largely on how accurate your IPA transcription is, because that is very much a scale. More often than not, what you see is a phonemic transcription rather than a phonetic one, which then does indeed require knowing how the actual phonemes are pronounced in a particular language. But with phonemic translation using the entire assortment of IPA diacritics, if your linguist is familiar with other languages that happen to have the same sounds, they could do a fairly decent take.
Do you know if there have been any of these kinds of recreations done on contemporary languages/dialects/accents, or I guess even created ones, to test the accuracy of these methods?
It sounds like something that should obviously have been done, but my naive googling isn't getting me anywhere so far.
Reminds me this video on what Shakespeare's original pronunciation sounded like: https://www.youtube.com/watch?v=gPlpphT7n9s