Could you get away with the following?

{
    my $shared_data;
    my $inner_sub;

    sub outer_sub {
        ...
        ... $inner_sub->(...) ...
        ...
    }

    $inner_sub = sub { ... $shared_data ... };
}
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That would get rid of the bug. (A named sub inside another sub in Perl is a bug in my eyes. The warning is too gentle.)

Note that having any $shared_data makes the function non-reentrant, but you can eliminate that problem by sharing data via parameters.

Update: no warnings 'closure'; with a comment explaining that all shared variables need to be exchanged by parameters or globals would also do the trick. But you're eliminating a very useful check.