perldiag says "(S) A severe warning (enabled by default)." which is probably part of the problem. But it is still easy to get around in a couple of different ways:

#!/usr/bin/perl -w
BEGIN { $SIG{__WARN__} = sub {
    warn @_ if $_[0] !~ /^Constant subroutine/;
} }
sub FOO() { 23 }
sub FOO() { 42 }
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Or

#!/usr/bin/perl -w
sub FOO() { 23 }
BEGIN { no warnings 'misc'; undef &FOO if ! @ARGV }
sub FOO() { 42 }
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I threw in the 'if ! @ARGV' just so one can see the warning produced or suppressed by the same code.