I'm not sure if it would quite be a farce, so much as a lot of lives and material being thrown down the drain ala the First World War because no one has a back-up plan for when don't go according to Plan A.
Either way, creative wheels are starting to turn, for better or worse...
Well, first we’d need a reason for 1930s Britain and America to go to war. There weren’t actually many reasons; neither power has any real reasons to pick a fight with the other. A lone incident – one side blundering across the other’s border - is unlikely to start an all-out ‘teach the blighters a lesson’ war. More likely, the diplomats would calm things down before they got out of hand.
Perhaps the post-WW1 diplomatic scene goes worse. The US tightens the screws on lending in a bid to get better battleship numbers than OTL. The UK backs Japan at the talks, rather than selling them out and ending the alliance. Britain thinks the US is trying to put a straightjacket on the Royal Navy, the US thinks Britain hopes to continue ruling the waves while waveing the rules. Japan plays on this, hoping to keep the UK onside. Then we wind up with the Irish Civil War, with the UK convinced the Irish are terrorists and the US sure they’re freedom fighters. The Irish cause all kinds of diplomatic ruptures when they draw weapons and supplies from the US, perhaps even mount terrorist attacks in Canada. The US would like to stop this, but public opinion is very pro-Irish and there’s little effective it can do. Things start to slide out of hand, with the UK sending troops to Canada to seal the border (which is pretty much impossible) and the US seeing them as a dagger pointed at the American heartlands; the UK also searching US shipping, as in 1812, and provoking massive bad feeling.
It's possible, in this timeline, Britain and Japan (and America) put more money into shipbuilding and land forces than OTL. Assuming they didn’t, the UK/Japanese alliance would have the edge, at least at first. On the other hand, the US would dominate the war on land. Fighting along the Canadian border would be costly, but the US would eventually push its way to the cities and take most of the country. (Halifax may not be so impregnable at this point.)
Meanwhile, the UK will have raided the coasts of America, taken the US installations on Cuba, captured the Philippines with Japan’s help and possibly taken Pearl Harbour as well. At that point, the UK runs out of viable targets. Raiding the US will hurt, but it won’t be fatal. The war might burn itself out at that point, with the US keeping Canada and the UK/Japan keeping its booty. (And all three sides having serious economic problems.) Or the war will stalemate until the US builds up the naval power to take the war to the UK/Japan.
If it continues, I imagine the US will mount major offences against British possessions in the Carabeen, as well as providing covert support to independence-minded activists across the British Empire. The UK might retaliate by supporting American blacks and Mexicans in hopes of keeping the US tied down at home. The fighting will go back and forth for a long time, but eventual US superiority will begin to tell and the British will be driven out of the region. At that point, I’d expect the UK to sue for peace.
If not, the war probably turns into Harry Harrison’s wet dream …
