If keeping regexes simple is one of your objectives, then another strategy would be to match all and then discard mis-aligned matches.

use strict;
use warnings;
my $str = 'AAA451BBB512CCC123DDD234EEE345FFF45';
while( $str =~ /((.)\2\2)/g ) {
        next unless $-[1] % 5 < 3;
        print "Found $1\n";
}
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