I'm thinking perhaps the error really means that there's "At least one Inline C function that doesn't bind to Perl".

I wish. It is what it really should be doing. But instead, I only get the warnings about AUTOLOAD. The latter appears to be something caused by Inline, it's probably introduced by how the subs and C code are connected.

But the former is what really worries me. If you try the code with no functions at all, like with an empty code block,

use Inline C => <<'END_OF_C_CODE';

END_OF_C_CODE
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then you get the warning, the first time you run it:

Warning. No Inline C functions bound to Perl
Check your C function definition(s) for Inline compatibility
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But no warning appears if there are both valid and invalid bindings. Ouch.