I can suppress all warnings using $^W

No, warnings overrides $^W/-w (regardless of their relative order).

use warnings;
BEGIN { $^W = 0; }
my $s = '' . undef();  # Use of uninitialized value
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Not just use warnings;. no warnings; overrides $^W/-w too (regardless of their relative order).

no warnings;
BEGIN { $^W = 1; }
my $s = '' . undef();  # [No warnings]
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In the context of your problem, $^W won't work if the sub was compiled with use warnings; in effect.

use strict;
use warnings;

sub test (&) {
    my $code = shift;
    local $^W = 0;
    $code->( undef );
}

test {
    print "@_";   # Use of uninitialized value
};
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That leaves you with $SIG{__WARN__}. You can catch the warnings and filter out the ones you don't want.

Update: Adjusted formatting for coherence. Added example to show that no warnings; overrides $^W too.