These tweets prompted a few interesting responses, but the discussion didn't really go anywhere. That's partly because I decided I'd rather make a real example than discuss hypotheticals.
Unfortunately, I didn't get as far as I would have liked. It turns out implementing a language is hard. But I did create a basic implementation, along with a Vim syntax file to enable syntax highlighting, and post it to GitHub as FlexVerb.
The readme contains 1,828 words. It starts like this:
FlexVerb is a barely-implemented programming language based on Latin and Ancient Greek. It is essentially a blog post, in executable code form, about an obscure linguistic quirk and its unexpected benefits.
Although I absolutely cannot recommend FlexVerb for production use, or even (in its current state) any programming tasks beyond the most utterly trivial, I can very confidently recommend reading its code — for anyone interested in parsing expression grammars, or a gentle introduction to language implementation and Vim syntax highlighting — and reading its readme, for anyone interested in programming language design or linguistics.
As far as I can tell, linguistics only concerns itself with the commonalities of human languages; I think my readme also implies a fairly compelling argument for the development of a linguistics which studies computer languages. Also, just to really push my eccentricity to new heights, I would love to see a linguistics which explored the commonalities between computer languages, human languages, bird calls, dolphin calls, primate communication, and the dancing of bees.
Update: Speaking of dancing, nearly everything I know about Romanian comes from dance music, and many of the things I say about Latin and Ancient Greek in FlexVerb's readme also apply to romance languages, especially ones whose classical origins are very easy to trace, like Romanian.