Unfortunately (though many think it a feature), your point 3. is correct. The declaration of %ENTITIES as a lexical gives it file scope only. Without some sub forming a closure by returning it or a reference to it, it is invisible outside the file.

There is a half-measure you can take. The functions exported by MathML::Entities need not be imported, and you can define your own %ENTITIES and functions like so:

package 'MathML::Entities::Approximate';
use MathML::Entities (); # explicit no-arg stops import
my %ENTITIES = (
    foo => '&x000042',
);

sub _convert2numbered {
    my $reference = shift;
    $reference =~ /^&([a-zA-Z0-9]+);$/;
    my $name = $1;
    # above copied from MathML::Entities
    return exists $ENTITIES{$name} ?
        $ENTITIES{$name} :
        MathML::Entities::_convert2numbered($reference);
}
# etc.
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Note that the _convert2numbered code copied from M::E is not very good. It assumes the match succeeded and blithely goes on to work with $1. Bad Practice. In its defense, such a match has already succeeded before this function is called, but that's a thin reed to hang on to if you want robust subclassible code.

Another choice would be to just copy the MathML::Entities code to your module and modify it there. That's not very different from the above code, and probably simpler.