You can use tr///c to replace all characters other than space and tab to something, and then use index to search for that character.

#!/usr/bin/perl
use warnings;
use strict;

sub regex_pos {
    my ($string) = @_;
    $string =~ /[^ \t]/g;
    return pos($string) - 1
}

sub tr_pos {
    my ($string) = @_;
    (my $tr = $string) =~ tr/ \t/!!/c;
    return index $tr, '!'
}

for my $s ("  \t x  ", "\t\t\t\t\t\t                  \xff...\t ") {
    regex_pos($s) == tr_pos($s) or die;
}

use Benchmark qw{ cmpthese };

my $s = ' ' x 200 . "\t" x 200 . "\x01" . " " x 200;
cmpthese(-3, {
    regex => sub { regex_pos($s) },
    tr    => sub { tr_pos($s)    },
});
[download]

On my machine, it seems about 3 times faster than regex in 5.26.1:

          Rate regex    tr
regex 316530/s    --  -67%
tr    954355/s  202%    --
[download]

but only marginally faster in blead perl:

          Rate regex    tr
regex 896519/s    --   -6%
tr    954359/s    6%    --
[download]