Hematohidrosis can be brought about by extreme stress and, yes, we know that our Lord sweated blood. Luke the physician describes it. However, I agree with DonaldC that it was the human nature in Jesus, fully man and fully God, that wanted to avoid the cross.
That would contradict his often expressed willingness and set determination to see this through. I understand your position is the one shared by most --- it just doesn't comport. To me that view is just totally foreign for the Lord. I just don't "buy it." He knew the Cross was necessary so why would he pray otherwise. However, the Lord may have been utterly caught off guard by the near heart-attack of hemohidrosis.
It was here that Jesus agonized in prayer over what was to occur. It is significant that this is the only place in the KJV where the word "agony" is mentioned (Strong's concordance). The Greek word for agony means to be "engaged in combat" (Pink) Jesus agonizes over what He is to go through, feeling that He is at the point of death.(Mark14:34) Yet He prays, "Not my will, but thine be done." GETHSEMANE--
Of medical significance is that Luke mentions Him as having sweat like blood. The medical term for this, "hemohidrosis" or "hematidrosis" has been seen in patients who have experienced, extreme stress or shock to their systems. (Edwards) The capillaries around the sweat pores become fragile and leak blood into the sweat. A case history is recorded in which a young girl who had a fear of air raids in WW1 developed the condition after a gas explosion occurred in the house next door (Scott). Another report mentions a nun who, as she was threatened with death by the swords of the enemy soldiers," was so terrified that she bled from every part of her body and died of hemorrhage in the sight of her assailants" (Grafenberg).