Ask Your DM. You have a number of questions here which don't have an official answer in the rulebook:
- How hard is it to put a bunch of lead on a canoe? How much does that cost? Can you carry the canoe when it's not underwater and full of air?
- How much air can you bring underwater, if you put it in a canoe and apply lead to balance the buoyancy?
- Can you breathe from that air long enough to walk a distance undersea and then ritual-cast Tiny Hut?
- How hard is it to breathe out of a small margin of air in the top of an inverted canoe?
- How hard is it to keep the canoe stable, so that the air doesn't spill out? If the air does spill out, how hard is it to get the suddenly-extremely-heavy canoe back to an air source?
- Are you planning to exhale into the canoe air, making it full of carbon dioxide and bad to breathe? Or are you planning to exhale over the side of the canoe, so you get less and less air in the canoe over time and it gets less and less buoyant?
- Leomund's Tiny Hut says it stays dry "regardless of weather" but it doesn't say that it "generates air if you cast it underwater". It sounds like you're reading "underwater" as a type of weather, and you're arguing that in order to stay dry the spell must be capable of creating breathable air. Does this work?
- Can the wizard's spellbook or warlock's tome tolerate being swum around with and used to spellcast in a canoe?
- If you get attacked while carrying this canoe around, are you basically just screwed, or what?
We could give you our best guesses at the answers (mostly, this seems unlikely to work very well), but these are the sorts of questions where your DM has to make a ruling, and anything we say would get overruled by your DM anyway.
(We can do some math regarding breathing: humans breathe "5-8 liters" of air per minute when not moving, or "12 liters" of air for light exertion. Multiply by two people, and assume you want to travel nine minutes undersea before spending 11 minutes casting the spell, so you need at least 400 liters of air. 400 liters of air displaces 900 pounds of water, so your canoe needs to weigh at least that much to balance it out.)
When your DM makes a ruling, they will consider a number of questions, including (1) whether this is practical, (2) whether it will advance the story, and (3) whether there's some other way you could search for this shipwreck that you should be doing instead.
There are easier ways to do this.
There are lots of spells that would work better than Leomund's Tiny Hut for exploring underwater. In particular, please consider buying, learning, and casting Water Breathing which is the same level as Tiny Hut. Alter Self is also good.
When your DM asked you to search for a shipwreck, the method they probably had in mind was that you'd just take a boat out on the sea and go diving periodically. (Or perhaps you'd make deals with some merfolk for the location?)