So in our world in the early renaissance period there was in fact one of the last great alchemist, for he wasn't yet a scientist, did in fact build submarines that did in fact travel underwater for some time.
Cornelius Drebbel for King James I of England in 1620's built submarines and Robert Boyle, one of the first scientists very highly regarded hims and investigated the subject:
Now that for which I mention this story is, that having had the curiosity and opportunity to make particular enquiries among the relations of Drebbel and especially of an ingenius Physician (Dr Kuffler) that marry’d his daughter concerning the grounds upon which he conceived it feasible to make men unaccustomed to continue so long under water without suffocation, or (as the lastly mentioned person that went in the vessel affirms) without inconvenience, I was answered that Drebbel conceived, that ‘tis not the whole body of the Air, but a certain Quintessence (as Chymists speak) or spirituous part of it, that makes it fit for respiration, which being spent, the remaining grosser body or Carcase (if I may so call it) of the Air, is unable to cherish the vital flame residing in the heart; so that (for ought I oould gather) besides the mechanical contrivance of his vessel, he had a chemycal liquor which he accounted the chief secret of his submarine navigation. For when from time to time he perceived that the finer and purer part of the Air was consumed or over-clogged by the respirations and steams of those that went in his ship, he would by unstopping a vessel full of this liquor, speedily restore to the troubled Air such a proportion of vital parts as would make it again for a good while fit for respiration, whether by dissipating or precipitating the grosser exhalations or by some other intelligible way I must not now stay to examine."
See further discussion here:
Given that one of the theories as to how he did this was via containing Oxygen that he had cooked out of Saltpeter and that the Chinese knew about Saltpeter long before that time period than it seems entirely reasonable to have there be submarines previous to that point in time back in the middle ages a little ways. The ability to have the man power and ship building to construct the vessels may be more of a limiting factor, as the Spanish Armada was sunk in 1588 by what was largely the English fishing and merchant fleet to the point that there were laws dictating the eating of fish.
I actually think that if you are wanting a fixed point to point navigation in a current then you may wish to consider having them use some sort of pully system; this would greatly increase the ease of navigating and put the power as being perhaps teams of oxen on either side so that the amount the vessel could hold would be increased. I am thinking that a chain system would actually be more reliable and cheaper for the day than anything involving rope as the links can be inspected and fixed by a blacksmith rather than having to have a rope walk and you are dealing with being subjected to water nearly constantly.
Drebbel's submarine held 16 people, I assume that having breathable air may have been the limiting factor. That gives something like 3000 pounds as the carrying capacity; you could probably assume that it is able to hold quite a bit more in terms of cargo carrying capacity than that if it were designed to carry cargo as cargo wouldn't necessarily need "a certain Quintessence (as Chymists speak) or spirituous part of" Air.