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?
I don't know "Dogen's wholehearted way" but I do know memory is needed to make perception of being. Memory is collected knowledge and experiences from the past. It makes YOU recognize, label, discriminate, it builds YOUR conditions to act or react and is unique for every living being. It all comes with a thought. Some are true, some are false. The truth is in yourself, you only have to meditate/pray to find it. Clear yourself from all negative energy and reach a higher state of consciousness, your memory then becomes irrelevant. Feel and be, don't think. Good luck!
Memory does not appear in the translated Bendowa
However, if we consider that memory might be defined as "held thought", then this passage from the Bendowa may be relevant:
All Buddhas continually dwell within this state, with None holding onto any of Their thoughts or perceptions, regardless of whatever may arise, whereas the great mass of sentient beings perpetually make use of what is within this state, but without their being fully awake to any situation
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.