I got it. It is quite simple. You have to do an 'imatch' operation in the query parameter. The code looks like this:
my $my_matches = $_my_table_class_mgr->get_objects(
query => [
my_field => {imatch => $my_value},
],
);
So, very simple solution.
I figured this out by studying the Rose::DB::Object::QueryBuilder
documentation here. Look under the part where it says 'Comparisons:' and then, just beneath that, where it says '"OP" can be any of the following:'.
Thank you to anyone who might have considered helping with this. I hope it helps a future newbie to Rose::DB::Object.