How to route to specific, of multiple datacenters, with a single domain
Proxying is gonna cause you some issues when someone takes a trip to China and tries to access data in the USA, depending on what your app does they could cause you some $$$ issues when your Chinese POP is sending/receiving all of their traffic each way.

Instead I suggest:

If your API is authenticated and you're not going to replicate any of the data and you're using AWS then I'd use lambda@edge to do what you need.

You can use GEO-DNS to route to a lambda function which handles authentication. This lambda function can also hold a record of where users need to go and issue a redirect to the correct endpoint.

This does mean that you'll still need domains like api-region3.company.com but it will allow api.company.com to perform the task of being universal. It's down to the client application then to hold on to that URL? At least using redirects instead of proxies should help with network traffic.