My name is Kameyama Junshi. I think there are two things to think about in response to this question.
One is whether it is OK to recite the Heart Sutra. As stated by the person who asked the question, the Heart Sutra is not sung in the Jodo Shinshu sect. This is because according to the teachings described in the Heart Sutra, the Jodo Shinshu sect is in the position that we cannot become Buddha. So what becomes an issue is whether the Heart Sutra should not be recited in the Jodo Shinshu sect, but the basic idea of the Jodo Shinshu sect does not say “if you do..., you won't be saved.” However, contrary to that, there is no teaching that “it doesn't matter if you do...” For example, if you take “lying” as an example, according to the teachings of the Jodo Shinshu sect, it's not “telling a lie doesn't save you,” nor “it's OK to tell a lie.” Amida Nyorai is a Buddha who saves any person. Therefore, there is no such thing as “if you do... you can't be saved.” However, Amida Nyorai reveals how persistent we are, and how we are incapable of doing anything good. There is no way for people who have such awareness to think “I don't mind doing it.” Therefore, I believe that “you must not recite the Heart Sutra” or “you can recite it” are not the teachings of the Jodo Shinshu sect, and simply “don't recite” is the correct answer. If so, in the case of this question, how should I think about it? My answer to this is linked to the second thing you should think about next.
The second thing to think about is, if you don't recite the Heart Sutra in terms of doctrine, is it OK to make a pilgrimage to Shikoku in the first place.
Below are my answers to these two things to think about.
The Shikoku pilgrimage and the accompanying chanting of the Heart Sutra may be good if you are in a position that it is different from your own belief in the Jodo Shinshu sect. If you think of the Shikoku pilgrimage as something on the same level as the teachings of the Jodo Shinshu sect (in other words, a different form of your own faith), I think this is an incorrect way of thinking.
These are my answers, but these are just my own thoughts.