That's how I'd solve the problem. The added complexity isn't even that high, give the application an endpoint to push an attachment into a distributed object store of your choice, submit a message with a reference to the object and persist it the moment the chat message was sent. This could be done with mere bytes for the message itself and some very dumb anycast-to-s3 services in different data centers.
I'm sure I'm skipping over tons of complexity here (HTTP keepalives binding clients to a single attachment host for example) because I'm no chat app developer, but the theoretical complexity is still relatively low.
