I know you are hoping it will escape people's notice but Halifax Nova Scotia is actually counted as being part of Canada now but was not then so I take it you know there are British regulars at Halifax in the period in question?
The fact remains as has been pointed out the US Army mustered some twenty five thousand troops total. Trying to overrun a British Imperial possession with some thirty thousand men was to be tried within a few years. It failed then. In additions but the passage time of the British Expeditionary Corps that would be sent to Canada would be shorter than it was for South Africa.
However as you say President Cleveland must persuade Congress to vote for a war of aggression. A hard sell. Likely to tip off British? Yes. Less risky for Cleveland personally than launching his army off ill prepared and lacking in supplies into Canada on Presidential authority alone though, especially when the bad news starts trickling back home.
Our whole argument (Or at least mine) assumes that Saphroneth's insane plan of bombarding US cities comes to pass. In that case the gloves come off.