That's probably a question for a lawyer, but I would not be surprised at an interpretation that a proxy that just mirrors the API of the thing it proxies doesn't insulate you from license compliance, in the same way that a library that just wraps a GPL library doesn't insulate you from license compliance. The question is whether the user is interacting with the AGPL product - if you're talking to software via a proxy, you'd likely say you're interacting with it, but if you're talking to some other software that happens to use that software, are you really interacting with it?

I guess the weird case is that when I'm using the Instagram app, I wouldn't say I'm personally interacting with even the Instagram front-end servers (the way I am in a browser), I'm just interacting with the app which happens to use the servers. And that does sound like not what the license authors would like.