Is sleeping through the care of a sick parent and not waking him for Fajr prayer or to prompt him with the Shahada and water, while also causing his blood pressure not to be measured, considered a dereliction of duty that warrants a guilty conscience and fear of causing his death?

There is no sin upon you, nor is there any liability for what you suspect was a result of your negligence, because the fundamental principle is the presumption of innocence, and a person is only accountable based on certainty or the predominant likelihood. If you doubt and are not certain that your father's death was due to your negligence, then nothing is required of you, based on the saying of Ibn Hazm: "And if he did not die from her action, then there is nothing upon her in that, or no blood money at all. If she doubts whether he died from her action or from other than her action, then there is no blood money in that, nor expiation, because we are certain of her innocence from his blood, and then we are in doubt whether he died from her action or not. And properties are inviolable except with certainty, and expiation is a religious obligation, and religious obligations are only binding by a text or consensus. Thus, it is not permissible to obligate her to a penalty, nor to fasting, nor to obligate her kin to blood money based on false conjecture."