>>110061913In reality, ambassadors are appointed by their own governments but only accepted by countries which want to accept them. It's not usually the case that any country would want to send a contentious pick to a country they want to have any kind of diplomatic relationship with - in fact beyond the big-name appointments, most embassies are just a suburban house with an office and a few staff, many of them local hires. This is because you only send an ambassador because you trust the host nation to a certain degree, and beyond that you take steps to protect your diplomatic staff appropriately.
Appointing a "certifiably-insane mass-murdering psychopath super-villain" of any nationality would mean that a) they had likely given up claim to their original nationality/citizenship (because they would be acting as your ambassador - in the case of US citizenship I'm pretty sure that's an automatic loss of US citizenship when you act as ambassador for another country to the US) and b) the appointment would be declined. Host nations aren't bound by any law to accept whatever whacko you send as ambassador - diplomatic immunity is something offered by the host nation, not something bestowed like a supernatural shield by the ambassador's own government.
Frankly, with the Joker, what would really happen in reality is that he'd be accepted, then the moment he entered the US his immunity would be revoked, a warrant would be served, and he'd be arrested etc etc.
But in reality, supervillians like the Joker don't exist, so it's not a very good example of anything.