This would be sufficient:

BEGIN {
    if( eval "use fakemodule; 1;") {
        print STDERR "Loaded fakemodule";
    }
    else {
        print STDERR "Couldn't load fakemodule.";
    }
}
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Don't put BEGIN blocks inside a subroutine, they won't do what you want.

Don't print in the BEGIN block, it will come before the Content-type header

Calling a subroutine with the '&a' syntax has the following effects, from perlsub:

• &NAME(LIST); # Circumvent prototypes.
  
• &NAME; # Makes current @_ visible to called subroutine.

In your code you used the &a; form, which would make the current @_ visible to the called subroutine. For example:

first(1,2,3);

sub first {
  &second;
}

sub second {
    print for @_;
}

# Output is 123
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Although it isn't relevant for this particular case it is still good to be aware of the prototype circumvention behaviour. Here is an example:

use strict;
use warnings;
use Data::Dumper;

sub pro(\@) {
    print Dumper \@_;
}

my @list = (1,2,3);
pro(@list);
&pro(@list);
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Output:

$VAR1 = [
          [
            1,
            2,
            3
          ]
        ];
$VAR1 = [
          1,
          2,
          3
        ];
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