Britain managed to send a total of several hundred thousand troops to South Africa during the Boer War, without facing undue logistical difficulties or leaving the rest of its empire open to attack.
The population of the United States in 1890 was 62,622,250.
https://www2.census.gov/prod2/statcomp/documents/1891-03.pdf (page 249)
If America is sufficiently determined, it would take way more than several hundred thousand troops for Britain to prevail. The winner would be determined by political will. I don't see the United States starting such a war and then giving up, because the circumstances that would make Americans likely to give up would also make Americans less likely to start a war in the first place. If Britain starts the war, then Americans definitely won't be giving up, no matter how much Anglophilia there was prior to the war. You need a situation where Britain is determined to win and USA isn't. I don't see a war arising under those circumstance. If America isn't determined to win, then America's not going to start a war. If Britain starts the war, then no matter how much Americans wanted to avoid the war, the public would not tolerate giving up. The Royal Navy could hold the West Indies, Newfoundland, and PEI, but if there's an Anglo-American War in the 1890s, mainland Canada would be annexed by the United States.
If the war breaks out quickly, the US' best option would be to occupy as much of Canada as as possible before the British can send serious reinforcements, and then dig in and use the occupied territory as a bargaining chip to try and get concessions elsewhere.
If there's a longer build-up, then assuming that the British can put a bigger initial force in Canada, the best option would be to dig in along the border and try and inflict heavy enough casualties on the attacking Brits so that their army is weakened enough for a counter-attack.
What concessions? There's the tripartite condominium in Samoa, but I doubt Washington would consider it worth more than mainland Canada.
I think people are over-estimating US coastal defences of the period. If one
believes Wikipedia 1890 is probably at or about the nadir of how bad they got, the old post-Civil War fortifications have been neglected, the report pointing out how bad they are has been written, but no actual work has started on building new modern ones.
If the US gets a few years notice of the war and starts a crash programme, they should be able to get something in place as they have the Endicott report to guide them. OTL took a decade or so to get 29 'major' locations defended and that was with a bit of rush during the Spanish-American War and the immediate aftermath, so it will have to be years as this is not a quick process even if you give it a high priority. But if they get that time, then yes British amphibious attacks would not go well or they would probably not be launched at all.
Again how are they going to sustainably project force? I never said they couldn't launch an amphibious attack. I said they couldn't launch them fast enough over a wide enough area to occupy the bulk of the US Army in a game of whack-a-mole.
But if the US only a few months notice, or it is the 'surprise' war option in the OP, then Britian is not launching an attack against '1890 technology', they are attacking some badly degraded and obsolete forts with un-trained garrisons and obsolete guns, which the US' government has already declared are utterly vulnerable to 'modern' ships.
I don't know where you get the idea that the US Army was "untrained" in 1890, but it's not true. If you're thinking about Brits training Americans in WWI, that was because new soldiers had to be trained, not because of any American practice of not training people in the army. You can have Britain win a war against the United States, but not in North America in the 1890s.
The problem is the logistics are pretty shitty in those parts of Canada and vulnerable to being pushed out by better-supplied British cavalry.
You don't need to hold territory for long to tear up railways.
If the US public is united behind the war and happy to take losses, then a large force can be sent to attack Canada and the RN raids might just harden public opinion. But if the war is not popular, if the state militias are deployed to protect their home state and not sent to federal control, then things are different.
Keeping the state militias out of federal control in this situation would require a POD before the ACW, probably one during or before the war of 1812.
The US regular army is small and will remain small for a long time if it doesn't get reinforcement from the states, with only the pre-war regulars and hurriedly trained and equipped raw troops the US will not be doing much of anything in the short term.
The trouble is the only way the United States enters a war against Britain without first enlarging the army is if Britain starts it. At that point the American public would not accept anything short of completely expelling Britain from mainland North America. Anything less would just be seen as leaving Britain with a base of operations from which to do it again.