It is not obvious what explains the divergence of reports, and this does have implications for the broader question of evaluating data and claims about languages. Methodologically, the first step would be to establish a fact about the modern language, for example that there is a word pronounced [mosir] meaning "X" ("every? land?"). Given the current state of the language and its geographical distribution, you would want to verify that the word is (or is not) uniformly pronounced [mosir], by which I mean "the narrow phonetic transcription is unambiguously [mosir]", and not "the phonemic analysis currently assigned to the word is /mosir/". I don't know if that much has been clearly established (I don't do Ainu). To the extent that there are materials for now-extinct dialects of the language, we would want to compare the form of suspect words across sources.
In the realm of comparing sources, one would want to look at competing transcriptions that dated back to roughly the period when Batchelor wrote, such as Dobrotvorsky's 1875 Ainsko-Russkiŭ Slovar [sic.]. I would not be surprised if this has already been done in Refsing's work Early European writings on the Ainu language. Batchelor gives some reason to consider the hypothesis that there are dialect differences. In The language, mythology, and geographical nomenclature of Japan viewed in the light of Aino studies starting p. 77, he gives details of the phonology, noting for example that "erum" is pronounced eremū, érem, erúm, erum, endrúm depending on location, illustrating the claim that there is "considerable uniformity" but "many words are variously pronounced in different villages and districts". He also notes that "There is a great tendency all over the country to confound the simple letter 's' with the combination 'sh'" (a propos the mosir ~ moshiri issue).
I will add anecdotally that I occasionally face problems correctly parsing Norwegian words with r+labial, such as Skærvøy, orm where I hear a bit of a vowel after the rhotic – some people pronounce the tap with enough vigor that it sounds like a vowel. If the issue is limited to final r, then maybe one can entertain the hypothesis that he mis-analyzed the final vocoid as being an actual vowel. He does observe that "ra, ro, and ru become n before n and t" and and notes a similar process affecting ro before t, chi. It is not plausible to maintain that he just imposed Japanese syllable structure on Ainu, since he give many examples of coda obstruents. I would search for examples provided by him that indicate a contrast between final r and r with a copy vowel or a contrastive vowel.
It is important to recall that the practice of removing redundant information from transcriptions is a modern theoretical practice. If you look at field work based records of that era, such as early Handbook of American Indian languages sketches (Bogoraz "Chukchi" etc.) you will notice a lot of phonetic detail that a modern work would suppress. Although Batchelor does not employ HAIL-style micro-phonetic transcription practices, it is plausible that he simply adopted a different theory of how to "phonemicize" certain sequences.