Buying Medium-hi quality Shabu-shabu meat in L.A?
Off of a random google JP search, the most popular cuts of beef for shabu-shabu are sirloin, chuck roll(kata rosu in japanese), rib eye and short rib. For pork it seems to come from the pork chop.
If you go to Nijiya they should say what kind of cut the shabu shabu meat is.
Both Mitsuwa and Marukai carry Snake River Farms shabu-shabu cuts. Mitsuwa did carry beef imported from Japan but they didn't have it last time I checked even though they had the signage. Mitsuwa also carries Certified Angus Beef (CAB) Prime, and Marukai carries Prime sourced from Harris Ranch.
Alternatively you could go to any good butcher and buy their meat and slice it up thin yourself.
It's not really a "cut" of meat, more like how it's sliced....
For the best deals, Try Korean markets. In Koreatown, I'm partial to Assi. Galeria and Plaza market are also good. There are also a few stand alone butchers... but they tend to specialize in things like Angus beef and can be a bit pricier... There are also Korean Markets in Torrance and Walnut...