Remember that post-Soviet Russia was a huge mess too. What benefit does a small Russian-majority successor state derive from joining Yeltsin's Russia? Russia then wasn't in a position to redistribute. And Russia already maintains pretty liberal laws towards citizens of ex-Soviet states. (All have been entitled to visa-free travel or resettlement in Russia - I think there are very lax rules about acquiring Russian citizenship for them too.) If the residents of an independent Kaliningrad (which might rename itself something like Kyonigsberg) or Karelia want easy access to Russia, they have it. They don't need to join Russia to obtain these benefits - in the process, they'd lose local control, upsetting not only local leaders but potentially also locals themselves.
It's also not at all difficult to Kaliningrad or Karelia for example cozying up to the EU. (This was the '90s, remember! Not like today when there's a lot more cynicism about Europe.) Europe, for that matter, would be happy to take them - even viewing them as a model state for Russia. And not at all difficult to imagine Kaliningrad residents especially thinking of themselves differently than other Russians - more European, more Western, Russian but not Russian.
The problem, like I pointed out above, was that Karelia never was a
real SSR. In any TL where the Stalinist period goes roughly as it did IOTL, it will be a province made of a dwindling ethnic Karelian population and a lot of Russians, many of whom have moved to the area since the war and have no real commitment to Karelia as a state-like entity. There was no Karelian national identity. All the other SSRs arguably had their own nationalisms, even if suppressed in Soviet times. The Karelian local identity, if there was one, was merely provincial and most people identified as just Russians and/or Soviet citizens in 45-90. Remember as well that Karelia, as a border area, had and has a lot of Russian military and other state personnel and officials loyal to Moscow rather than a local leadership, and unlike in the Baltics, say, there would be no local movement to try and expel these troops as they are not seen as "foreign". Also, Karelia is next door to the second city of the USSR, Leningrad, forming much of its hinterland. Any attempt to break away from Russia would be noticed quickly in Leningrad and there would be measures to stop those pipe dreams.
Also, Europe being happy to take them? The closest European nations, Finland and Sweden, would be very critical towards ideas of Karelian independence. Remember that they were careful with the Baltics as well, in the early days. And unlike the Baltics, of course, Karelia has no history of independence from Russia. It thus would have far less legitimacy than an Estonia, say, for its attempts to break away from Russia.
In short, it is IMO hard to see realistic pushing or pulling factors for Karelian independence, like it is hard to see reasons why the Finno-Karelian SSR would have stayed a separate entity with a SSR status after the 60s. That said, Karelian independence is an intriguing idea. But it is one that IMO would require a POD or PODs in 1900-1920 at the latest to be realised, preferably even earlier.