Which is precisely the reason why the Hartford Convention never declared or outright supported secession - indeed, IIRC what I know about the Convention, the delegate selection process was designed in such a way to specifically exclude people who would push for secession right away. The last thing the Federalists wanted was to create more trouble than it needed. In that sense, whether or not it was intended, the Hartford Convention helped to save the Union in more ways than one. Now, if it was before the OTL Convention, then it would be a different story, although it would be more of a piecemeal operation and could involve Britain especially exploiting how ruinous the reinforced embargo would be (Dathi THorfinnson used
a similar plot point in his Canada-wank TL; if I wanted to work something like that into a New England secession TL, I'd start with Rhode Island first because of its weird relationship to the rest of the region, let alone the rest of the country (sorry, Oliver Hazard Perry) - last one in the Union, first one wanting to find an escape route). There could be an internal civil war here, viz. NH and VT versus the rest of New England, which may or may not be in the secessionists' favor.
All in all, any successful secession during the War of 1812 must take place
before the OTL Hartford Convention, IMO. OTOH, since secession would be a nasty affair, an independent New England would want to make it as soft as possible so that it doesn't upset the North American balance of power too much. For example, take New England's portion of the national debt. Even if the currency situation fluctuates towards New England's favor, I could easily see Beacon Hill being obstinate enough to insist on paying all of it in full to the agreed schedule, no more and no less, and even if it means long-term hardship.