Is the son/daughter's negligence in caring for his/her parent at the moment of their death, due to extreme exhaustion, forgivable, and what must he/she do to atone for that?

What you did was not disobedience to your mother. It seems you were dutiful to her and keen on pleasing her, which bodes well. Even if some shortcoming occurred from you, Allah forgives slips and unintended actions if the intention was righteous and dutiful, as He (the Exalted) says: "Your Lord is most knowing of what is within yourselves. If you should be righteous, then indeed He is ever Forgiving to the oft-returning [to Him]."