Does Dogen's wholehearted way say anything about the status of memory? Seems to me that memory is notoriously fallible, whatever we think about the nature of the self. I have a copy, but rather than reread it, I thought to ask, whether Dogen's wholehearted way, or right effort in general, from moment to moment, involves relying only on those memories of immediate relevance?
If you are referring to the intellectual capability of memory (隨念智), then this is not mentioned in Bendowa.

The whole-hearted way is, as you say, awareness from moment to moment. According to Zen, it is the experiential reality, not the memory or interpretation thereof (even if the memory is one of realization). The memory of a previous realization cannot help you. It will in fact hinder you if it arises. This is well explained in Mumonkan case 22:

Ananda asked Kashyapa, "The World-honored One gave you the golden robe; did he give you anything else.
"Ananda!" cried Kashyapa.
"Yes, sir!" answered Ananda.
"Knock down the flagpole at the gate," said Kashyapa.