I honestly don't mind people saying things like "carry-okey". I am 100% capable of saying カラオケ with a flawless heiban pitch accent, with vowel and consonant qualities that would totally shock natives and make them do triple backflips out of excitement... But I'm also not going to just say "I was home alone so I did some カ{ラ|}{オ|}{ケ|____} to my favorite songs while no-one could hear me." — the pronunciation of "karaoke" going to be anglicized in one way or another, and English does indeed have a limited set of phonemes and some number of phonotactic restrictions and so forth. Relevant for "carry-okey" is diaeresis and the checked vs free vowels.

Ultimately I don't think that "carry-okey" is any more "criminal" a pronunciation than Japanese people themselves saying, well, kara-ōkesʉtora — the Japanese tendency to epenthesize extra vowels in loan words to satisfy that language's phonotactic restrictions is literally optimality theory 101 type stuff. But we place higher expectations on Americans to say things "correctly" basically just because they're Americans: this judgment is divorced from all that boring academic linguistics stuff, and is more just about preconceptions and prejudices and engaging in the social activity of dunking on how dumb Americans are...

...I dunno, I grew up speaking two languages myself, English and Norwegian. I'm already very used to Americanizing my own name and place names in my local area and things like that, to my relatives who only speak English, as well as Norwegianizing the names of relatives and place names in the United States. My mom actually argues with me on occasion that I Americanize my surname "the wrong way", because her pronunciation of our surname tries to approximate the original Norwegian pronunciation as much as possible, whereas my pronunciation is a lot more orthographic and "casual". And I just tell mom when that topic comes up that I'm the one of us two who actually speaks Norwegian as a first language, so if I want to use a diphthong where she uses a monophthong or a schwa where she uses a long vowel, just because I think that sounds better, then that's my prerogative: it doesn't come from me being unable to pronounce my own surname in its original language, obviously, and it's not a matter of dumbing it down for foreigners, either, it simply sounds better to me and that's all there is to it... Well, "all there is to it", I do think that there is some underlying baggage about things like language politics or national identity that underpins the whole "how do we say our own surname?" debate, when it's between an immigrant and an immigrant's child.

But regardless, this doesn't mean that I don't also cringe at the ways other Anglophones say a lot of Norwegian place names, or for that matter how other Norwegians say a lot of English loanwords, and in these situations I'll think, "They should use my pronunciation instead! Mine is way better!"... But ultimately I know that this is all very silly, right? Because I know from experience how awkward it can be to try to convert a word from one sound system to another, and I'm sure that this goes doubly for those who grew up monolingual. So the ways in which other people try to adapt a word or name into another language isn't of any less value, fundamentally, it's just... different from what I personally would've done. My pronunciations are just one approach among many possibilities.

The one who can make peace with this fact should make peace with it.