The Hartford Convention was never about succession. It was about getting the Federal Government to take New England interests into account. The Republicans were more concerned with the interests of middle Atlantic, and Southern States.
Indeed, when the main reason given for a war is about shipping, then the region that has all the ships, as well as bordering the area that would see warfare, would naturally want to have a say in the matter. Little old ladies don’t want help crossing the street when the pharmacy is on the side they are already on.
It is a bit more complicated than that. The Hartford Convention was a
compromise convention, to find a new common ground with the Madison Administration so that the war could end favorably for the Administration without further hurting the New English economy and not permanently harming Anglo-American relations so that the NE fishing economic monopoly did not falter. IOTL the news of the treaty and New Orleans came
after the convention had sent its demands to Washington making them look like fools otl. However secession was not ruled out. George Cabot, Benjamin West, Chauncey Goodrich, Daniel Lyman and William Hall, the respective leaders of Massachusetts, New Hampshire, Connecticut, Rhode Island and Vermont Delegations agreed that if their demands were not met, then a second convention would be about secession. In a scenario where the War of 1812 is stretched into late 1815 New English Secession becomes a real threat to the USA. As it was, Massachusetts was basically conducting its own foreign policy iotl after exasperation with the federal government instituting the Maine Ceasefire on its own accord. Vermont and New Hampshire's Senates were discussing same measures iotl when the news of the treaty came.