but how do I get to the decimal cents in the first place?
Assuming you start with a string, you have to avoid triggering perl's implied parse to a floating point number.
my $input= "2.26";
my ($whole, $frac)= split /\./, $input;
$frac //= '';
length $frac < 2 or die "would lose precision on .$frac";
$frac .= '0'x(2 - length($frac));
my $cents= "$whole$frac";
Yes, it's ugly. I'm sure there is some clever way to do this with a regex that runs faster, and would be even uglier. But, you can just stick that in a function named "parse_cents" or "parse_decimal($str,$scale)" and hide the details.
Another option is to use Math::BigRat, which will handle all arbitrary cases, but then the performance of all the math you do on the numbers will be a little slower.