I don't think the /m/ of mientras implies a great deal about the pronunciation of Latin -m beyond what is already known from other sources.
Mientras is clearly not just the regular outcome of applying Latin-to-Spanish sound changes to Latin dum interim. If so, it would not end in -ientras; Latin interim should go to entre with no final s and no diphthongization of the tonic vowel. The presence of these irregularities makes me think that /m/ could possibly be another irregular development in this word, rather than a sound that was preserved from a feature of Latin pronunciation.
It has possibly been analogically influenced by the adverbial suffix -mientre
In A History of the Spanish Language, Ralph Penny does give the usual etymology from dum interim > domientre > demientre > mientras, but says that the "diphthong results from the influence of the Old Spanish adverbial ending -mientre < -MENTE" (page 247).
The /r/ in the form -mientre possibly shows influence in the opposite direction, from domientre (Penny 132). (However, an alternative etymology of -mientre traces the r to the Latin adverb-forming suffix -ter: Syntax and Affixation: The Evolution of MENTE in Latin and Romance, Keith E. Karlsson, §5.7.2.)
If the ie in mientras was caused by influence from -mientre, it doesn't seem too implausible to me to suppose that the /m/ might also be influenced by -mientre, through a development like [dõentre] > [domɛntre] or [don entre] (compare the [n] that developed in the monosyllables quien, con) > [domɛntre].
possible alternatives to the etymology from dum interim
While the etymology you give seems to be widely acknowledged, there is a paper from 2011 that presents other etymologies: "Nueva hipótesis sobre el origen del adverbio medieval 'm(i)entre'." Javier Mora García.
García's paper is in Spanish, so I don't understand the details of the arguments. THe first novel proposal gets rid of dum, although it looks implausible to me: en entre > enentre > de enentre > denentre, then, supposedly by an dissimilation of nasals, dementre. The second involves dum, just like the standard etymology.
However, the third etymology García proposes gives a regular explanation for the /m/ as not coming from Latin word-final -m, although I don't know if I'm convinced by it either. He suggests mientras could be ultimately from interim + -mente with alteration of the second element and then loss of the initial entre-. A form entrementre is attested in French, and entrementes is attested in Galician-Portuguese.
Ultimately, I don't get the sense that the standard etymology is convincingly displaced by any of these, but it seems like something to consider.
Latin final -m might have had [m] as a possible pronunciation, even if "mientras" by itself is weak evidence for it
For the reasons above, the /m/ in mientras doesn't seem very strong evidence in and of itself for supposing [m] in the pronunciation of Latin dum.
However, there are unrelated reasons to think [m] might have been at least a possible occasional pronunciation of word-final "m" before a vowel in Latin.
The most obvious is the spelling. Latin spelling is mostly phonetic, and the letter "m" definitely could represent [m] in other contexts in Latin words. Early on, when Latin spelling conventions were first formed, the sound was presumably felt to be similar to [m] (one fairly common suggestion is that word-final -m could be realized as a nasalized glide [w̃], which like [m] is a nasal and labial non-syllabic sound). And after Latin spelling had become conventionalized, the force of spelling pronunciation presumably could have introduced [m] into the speech of literate speakers.
The etymological source of Latin final m is reconstructed in Proto-European as *m, which is normally assumed to have been pronounced as [m]. One small piece of evidence that survived in Latin itself for [m] as the etymological value of final -m is the form of the verb comedō, whose etymology is typically given as from edō and the prepositional prefix corresponding to cum (which usually took the form co- before vowels).
Lastly, there are references in grammars to a fault of speech "mytacism" or "moetacism" that was in some way connected to the pronunciation of "m". It's not always clear what a grammarian means by it, but some cases seem to refer to pronouncing the word-final /m/ of accusative words as [m] before a vowel (Nyman 117).
The consonant [m] is not considered to have been the usual pronunciation of word-final Latin -m before a vowel because of the regular occurrence of elision in this context: we find singultantem animam et in Ovid pronounced as six syllables, long-long-long-short-short-long, which fits perfectly with [singultant‿anim‿et] and is not really compatible with any pronunciation where the -ms are [m]: [singultantm‿animm‿et] has an awkward cluster after the third syllable and looks like it would have a long fifth syllable.
A History of the Spanish Language. Ralph Penny. Second ed. 2002.
"Nueva hipótesis sobre el origen del adverbio medieval 'm(i)entre'." Javier Mora García. Anuario de Lingüística Hispánica, ISSN 0213-053X, Vol. 27, 2011, págs. 127-144
"Mytacism in Latin Phonology". Martti Nyman. Glotta 55. Bd., 1./2. H. (1977), pp. 111-120 Published by: Vandenhoeck & Ruprecht (GmbH & Co. KG). Accessed through Jstor.