With two near threats in the Far East it is highly unlikely that the Russians will want to add a third,
Can you really not conceive of any way the US and Russia could end up at war?
What if Russia and Japan come to an accord to divide up China between them and the US objects?
Or what if Russia and Germany form an alliance and Germany gets into a conflict with the US and calls on Russia for help?
Or what if Russia and France continue their existing alliance and France gets into a conflict with the US and calls on Russia for help?
Or what if Germany gets into a conflict with Russia (or the Soviet Union if the communists come to power) and calls on the US for help?
None of those scenarios are unthinkable and every one of them could lead to the US and Russia in a shooting war in the Pacific, and remember there doesn't seem to have been a Russo-Japanese War in TL-191, so the Russians are a lot stronger in the Pacific than they were IOTL. (They presumably still have Port Arthur and haven't lost their Pacific Fleet.)
But even if the Oracle of Delphi showed up and personally gave the US president a 100% prophecy guarantee straight from Apollo himself that the US will never be in a real shooting war with Russia from that moment to the end of time, having Alaska would still be of use to the US for keeping out the British and Japanese. And those nations using Alaskan waters to get to the US is not just a theoretical danger. It straight up happened in the First Great War. Here's the relevant quote from the novels.
The Great War: American Front said:
"He picked up snatches of their conversation-place names mostly. 'Kodiak... Prince Rupert... Victoria... Seatle.'
'Since he'd just been thinking wistfully of a cooler clime, he called after them: 'What about Seattle?'
The two men stopped. 'Nothing good," one of them answered. "The goddamn J*** have reinforced the limey flett off British Columbia.'
'You're right-that isn't good,' Sam agreed. The places they'd mentioned made sense to him now. 'They sailed up by way of Russian Alaska and then down along the west coast of Canada, did they?'
'That's what they did, all right, the bastards', the other sailor agreed unhappily. 'On account of it, the North Pacific Squadron can't hardly stick its nose out of Puget Sound.'"
-Great War: American Front, Chapter XIX
And it's not like it would take an enormous garrison to take and hold Alaska. Alaska was barely populated in the early 20th century and there's no reason to think that is any different in TL-191, so it's not going to be seething with rebellion. (IOTL in 1897 the US Army garrison in Alaska consisted of a grand total of 30 men.)
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TL-191 US will presumably want a slightly bigger garrison than 30 men, but a single regiment would be more than sufficient force to hold the territory.
So what exactly is the strategic logic in failing to take a territory which would help you plug a gap in your nation's defenses that the enemy exploited in the last war, and which would not require substantial resources on your part to take and hold?