The regex match inside the while condition stores its current position in pos($string) (see pos for detail), which the substitution resets, so on the next iteration it starts from the beginning again.

So if you insist on doing the matching and substitution in two different steps, you have to manually set pos($string) after the substitution:

use strict;
use warnings;
use 5.010;

my $string = "CATINTHEHATWITHABAT";
my $regex = qr{\wAT}i;

while ($string =~ m/($regex)/g){
    my $match = $1;
    my $start = $-[0];
    my $end = $+[0];
    my $hitpos = "$start-$end";
    my $lcmatch = lc($match);
    $string =~ s/$match/$lcmatch/g;
    # in the next iteration start over where we left off
    pos($string) = $end;
    say "$match found at $hitpos ";
}

say "string: $string";
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Since you put parenthesis around the regex, ${^MATCH} can be replaced by the shorter $1, and there's no need for the /p modifier.