As I understand it, the square riggers needed to run with the wind and generally couldn't beat into the wind very well. There have been numerous questions as to why Bounty went from a SE course to SW. A hurricane rotates counterclock wise. So initially, ship would have likely had winds from E or NE since it was ahead of the storm and to the west of the eye track. As the ship moved more east and south, eventually at some point it would be to the east of the eye track. At this point the winds would increasingly come from the SE, then S. So this would mean that ship was increasingly heading directly against the wind, which is an impossible situation for a square rigger. But if the ship could stay to the west of the eye track, the winds would increasingly come from NE, then N, which would be suitable for running before the wind on a SW course. So, unless he were to change to a more northerly course, the captain didn't have much choice except to turn SW. If the pumps had been pumping at their rated capacity, they still might have made it despite that they would have had to contend with the Gulf Stream, and the unique feature of this particular storm, wherein it was reported that the highest winds were in the SW quadrant. Once in this quadrant, the storm would be moving away from them after the eye passed to the east and north in any case.
Sometimes I am slow
If you are right about the wind direction and force I guess you nailed one of the things that have been discussed here at length and I only noticed now, I mean why the Captain did not manage to do what he said to the crew he was going to do before sailing.
If you are right he went SE as planned to give space to the hurricane to pass between him and land but soon discovered that the winds ahead of the Hurricane, frontal winds, were this time to strong to permit him to make any decent way. As you say these ships do not point well and the engines had not the power to make it against a strong wind and waves, not to mention that the ship should be pounding heavily.
So he went till where he thought it was possible and then with the ship not doing any significant speed and taking a lot of stress, decided to do the only thing he could, turn the boat to the better sailing position regarding the wind and waves, minimizing the stress on the boat and gaining speed.
That explains why nobody new very well to what course they were steering the ship: They were in survival mode and steering in a way to get the less possible stress on the boat, according with the wind and wave directions, vaguely SW.
Bad luck with the wind and wave direction forced them to those bad waters near that cape and even so they only capsized when the boat was full of water. The captain had put himself in a situation that he had no choice.
As NCC320 says, if the boat was in sound condition, not making so much water, if the pumps were working at their max rating speed, if they did not have lost and engine over an accident, if the crew was trained to serve the pups, if the set up of the pumps was correct, if they had a professional crew, they could have made it and even so it would be a risky situation that should be avoided at any cost by any Captain.
Just to see if NCC320 is right with this possibility: Has anybody any means to determine the overground speed the Bounty was making SE immediately before turning to SW?