I now see that this was answered on the next page of the book.
Harvey's "ounces" were today's ounces, and his drams were "fluid drams", each one eighth of an ounce. It isn't explained why he identified "half an ounce" with "three drams", but possibly he found "three eighths of an ounce" cumbersome and unnecessarily precise, he was imprecise about other figures, too, such as assuming a human heart rate of around 30 bpm instead of the normal 70 bpm.
I haven't found a precise description of Harvey's methodology for measuring stroke volume, but a museum webpage gives this description of his successor, Stephen Hales' method:
This scientifically interested layman undertook in Tordington (sic) in 1710, 53 years after the death of William Harvey (1578–1657), the first precise definition of the capacity of a heart. He bled a sheep to death and then led a gun-barrel from the neck vessels into the still-beating heart. Through this, he filled the hollow chambers with molten wax and then measured from the resultant cast the volume of the heartbeat and the minute-volume of the heart, which he calculated from the pulse-beat. Besides this, Stephen Hales was also the first, in 1727, to determine arterial blood pressure, when he measured the rise in a column of blood in a glass tube bound into an artery. (From the acceptance speech for the Nobel Prize in Medicine, given by Werner Forssmann in 1956.)
Regarding Harvey's methodology, I found is in this commentary:
He isolated parts of the heart; he ligated and divided arteries; he exerted pressure on veins on either side of the valves. His observations of dissected hearts showed that the valves in the heart allowed blood to flow in only one direction. Harvey measured the volume of the left ventricle and calculated that the amount of blood that passes through the heart of a man in an half hour and established that it was greater than the amount contained in the whole body
And this discussion of Harvey's results:
To make his calculation Harvey had to measure the pulse rate and the amount of blood that the heart ejects with each beat. Here, he tackled a tough problem; even today there is no widely accepted figure for cardiac output. The determination of cardiac output is most difficult, and the various procedures
for measuring it give results that vary as much as 25 per cent. The lowest figure, however-one dram, apothecaries weight-which Harvey used was only one-eighteenth of the lowest calculation accepted today.
So it seems that Harvey wasn't overly concerned with precision, and he used only crude methods of inspection to estimate the heart volume, meaning he was off by quite a bit, although that didn't really matter much for his arguments.