Honoring the recently deceased. Is it always appropriate to chant a Heart Sutra after an offering has been placed on the altar? Is there an occasion in which a Dai Hi Shu is best?

Concerning the sutras, which one you actually ready does not create a problem, because it is simply an expression of your honest mind. The Daihishu is not translated and still in the original Indian language, because it is a sutra not stressing the meaning but rather the chanting of it, whereas the Heart Sutra is stressing its inner wisdom based on emptiness. So either of these two are fine, but maybe the Heart Sutra may be more suitable.