More importantly, the US has to cede the south border of the St. Laurence, losing Ogdensburg, for instance.
Now Britain/Canada KNOWS the importance of the Lakes, AND has the whole path secure.
The problem is - how secure is secure? It's a very long, thin strip of land, which makes fortifying it difficult, and any troops you station on the south bank run the risk of being flanked and cut off from their line of retreat - or having to retreat across a river. I suspect as much as possible gets put on the north bank, simply because the Americans then have to cross a river to capture or destroy it. The only section that seems to really affect is the Long Sault/Barhart Island/Cornwall area, though.
IF it's a British Imperial project, they'll probably make sure that then current RN warships can sail through.
This is a problem, because in the early 19th century ships get a lot bigger quite quickly. Some illustrations:
HMS Lively (standard frigate of the Napoleonic wars, commissioned 1804): 47m length x 12m beam x 4.1m depth of hold
HMS Shannon (typical screw frigate, commissioned 1856): 87m length x 15m beam x 5.6m depth of hold
HMS Cormorant (typical ship-rigged sloop, with 24 ordered 1805-6): 33m x 9m x 2.74m depth of hold
HMS Racer (typical mid-Victorian sloop, ordered 1854): 46m x 9m x 5m depth of hold
HMS Swiftsure (typical 74-gun battleship, commissioned c.1804): 53m x 15m x 6.32m depth of hold
HMS Bulwark (last design of wooden battleship, c.1859): 77m x 18m x 8m draft
Data on merchant ships is harder to come by, but mail ships are certainly getting similarly bigger:
SS Great Western (launched 1837): 72m length x 18m width
RMS Persia (launched 1855): 121m length x 14m width
The risk - as with the original canals on the St Lawrence - is that the British build them small to start with and watch them become rapidly obsolete. The opportunity, of course, is that they manage to establish an early Panamax standard. But shipbuilding and metallurgy are advancing so rapidly that you might well end up creating a bottleneck.
If anything, Britain uses the money saved from the St. Lawrence canal (much cheaper than the Rideau) and builds the Trent-Severn Waterway.
I've always thought that the
ship canal between Lake Ontario at Toronto and Lake Huron at Wasaga Beach that was proposed on the late 1860s (though apparently surveyed in 1846) made more sense than the Trent-Severn waterway. It's difficult to justify the expense over and above the Lake St. Clair route, though, especially if Michigan isn't potentially enemy territory.