Then, whenever the API includes an id in a response, it adds the appropriate prefix. When receiving an id in a request, validate that the prefix is correct and then strip the prefix.
That gets you all the advantages of prefixed IDs but still keep all 128bits (or however many bits, you don't have to stick to UUIDs) for the actual id.
Or, to put it another way, there's no need to store the prefix in the column because it will be identical for all rows.
EDIT: this is not to knock your work - quite the opposite. If you do have a use case where you need rows in the same table to have a dynamic prefix, or the client takes the IDs and needs to put them in their own database, then your solution has a lot of advantages. I think what I'm getting at is that if you're using prefixes then it's a worthwhile discussion to be had about where you apply the prefix.