Tricky situation moving with a pet originally from USA now in Korea to Japan

After visiting Japan several times throughout my life I planned on moving to Japan however plans changed when I met my fiancé :
I adopted my cat in the USA prior to meeting my fiancé , I then moved to Korea for what I thought would be the rest of my life as I was engaged and my fiancés aging parents lived in Korea so we moved to be close to them. However my engagement ended due to my partner cheating ( obviously when I got engaged and moved with my cat I didn’t predict that would happen) , I have stayed in Korea for 2 more years after that doing my best to make it work however amongst a few things the air quality in Korea has made me very sick despite moving to a different area in Korea that was supposed to have better air quality. Anyways since I really have no reasons to continue stay in Korea , I reached out to an old professor of mine who offered me a job for January 2027 in Japan. The flight from Korea to Japan is very short (only about 3 hours) so I know my cat will fare well ( she also did well on the original long flight to Korea when I first moved here – she’s very healthy too) and I originally wanted to move to Japan before I agreed to move with my fiancé to Korea I agreed . I have about a year to prepare for the move with my cat and the trouble I’m running into is Japan doesnt really recognize vets from Korea … which makes getting stuff like the blood titer test and Government endorsed signatures similar / equivalent to the USDA approved vets I used when I originally moved to Korea to be a bit difficult. Theres also issues of Korea not being the country of origin for my cat . Bearing that in mind can anybody help / guide me with this process ?

Also yes I would not have moved my cat to Korea if I thought my engagement would end , I thought I would be married to this guy and he didn’t cheat until we moved to Korea so there was nothing I could have done to know. I am not moving to Japan to be with someone and I have visited Japan often both as a kid and as an adult now . I also have extended family in Japan.

by Possible-Biscotti393