
I'll try one last time to get past that Titanic sized blind spot you have... But you do realize the concept of defense and offence, right? If the British are INVADING, they are on the OFFENSE... right? They have to cross rugged terrain with little infrastructure to reach anything of value in the Union. Now the Union, in this instance are the ones being invaded so they are on the DEFENSE... they have to stop the Brits from capturing their cities and destroying industry and what not, since the Union is already there in those cities they can prepare defenses, right? Now, if the Brits are invading they need enough men to gain control of places even over the defenses... They need enough supplies to make it down to their target, overcome the defenders and hold that position long enough to either damage or destroy its value or force the Union to fight on their terms, right?
Now, you keep going on and on about the Union's manpower problems... the Union has a population of 22 million... In the OTL ACW the Union mobilized 2 million men over 4 years, and aside from CSA 2 offensives that didn't do much disruption, remained unmolested and on the offense throughout the war. They had war exhaustion because they were on the offense the whole time. Now they are on the defense. The British invasion is threatening their homes and cities... a much better motivation to get up and fight for the Union than esoteric national concepts and later slavery. The CSA mobilized a much larger percentage of its fighting population than the Union (15% vs 9% or thereabouts) Every 1% greater mobilization the Union is able to rally is 220,000 more men... and I seriously doubt the Union is so lacking in motivation that they couldn't raise their mobilization levels to face the Brits.
Now we have the British side of the invasion equation. If the Union is able to raise 200k men who are willing to defend then the Brits have a problem. Their hard core of trained men is 25k and those are far flung all over the Empire, the rest will be on the same level as the Union troops in training. Even by your best metrics the Union now has a 10:1 advantage, better than your 6:1 magic ratio which you seem to think is what one 'Spartan II' Brit is worth vs one Union soldier (I don't accept that at all but lets give you all benefit of the doubt here for the sake of argument). So now the Brits have to raise an army above what they got. They have to ship that army over the Atlantic, they have to base that army deep in Canada, they have to march that army over the Adirondacks to hit anything of value, and if they lose or are forced to implement a siege they have to stay at the end of a very long supply line while the Union is working off its internal rail and road networks...
Now, if we robotomized everyone, took out politics and simplified logistics and boiled it down to a game of Civilization the Brits could, theoretically, muster more men than the Union, they could put a million man army in Canada, march it down to NYC and take it after sustaining Nappy'esque casualties while building another million man army to replace the losses... but this isn't a game of Civ... and the Brits don't have the nofog cheat on to see exactly what they are going up against, all the while the Union has the home field advantage.