Given that a UUID identifier fits in a single cipher block, and the whole point is that these are unique by construction (no IV needed so long as that holds true), it seems like a single round of ECB-mode AES-128 would enable quickly converting between internal/external identifiers.
128 bits -> 128 bits
Yep have used an approach just like that, worked quite well if you have a strong pattern to easily translate from one to the other. Gives you an id with the right properties for internal use, efficient indexing etc, and in its encrypted form gives you the properties you want from an external identifier being unpredictable etc, all from one source id.
It is true that now your encryption key is now very long lived and effectively part of your public interface, but depending on your situation that could be an acceptable tradeoff, and there are quite a few pragmatic reasons why that might be true as has been described by other comments.
Edit: you can even do 64bit snowflakes internally to 128bit AES encrypted externally, doesn’t have to be 128-128 obvs
No need to encrypt, just store the external key in a table. Not that you're likely to change algorithms.