I don't know very much Lojban but (while I agree that the grammar is meant to be a realization of a formal logic), I'm a bit unsure of your specific example with "lo cmalu noltru":
First, I think "lo cmalu noltru" is a sumti (akin to a noun phrase) and not a complete sentence by itself.
Second, I'm not positive about the existential import of "lo". There are many reasons for this, but to take a kind of extreme example, the reform of the meaning of le and lo at https://mw.lojban.org/papri/BPFK_Section:_gadri#cmavo:_lo_.2... presents the example
lo pavyseljirna cu ranmi danlu gi'e simlu lo ka ge ce'u xirma gi lo pa jirna cu cpana lo mebri be ce'u
'Unicorns are mythical creatures that look like a horse with a horn coming out of their foreheads.'
This doesn't seem to imply that there exists X such that X satisfies the predicate pavyseljirna. Or to take another example, presumably Lojban can express using "lo" the notion that "Counterexamples to Fermat's Last Theorem were never discovered prior to 1993, despite extensive computer searches".
I do think that many Lojban utterances have formal existential import in many cases, and I don't know enough about Lojban to try to describe which ones. :-) (I seem to remember something about a bare gismu implying that the predicate is at some occasion true of something, like "tavli" by itself without arguments meaning something like 'speaking, it's a thing that happens at some point!'. But given the example of talking about unicorns, maybe even this is too strong somehow.)
Edit: ... by the way, what's the conlang with two speakers that you speak?