After some investigation and benchmarking, it looks like the best PIR protocol for this use case is YPIR+SP (from February). On a single compute- and network-constrained server, with users on constrained (and possibly metered) networks, this would amount to providing service to up to 1000 users while keeping latencies reasonable; by (quadratically) scaling the server(s) enough, that could become up to 100,000. That means this method of message routing could definitely work, although I look every day in case new protocols are published.
No, sorry, I haven’t uploaded anything yet, I’ve only coded the protocols and some benchmark code. The idea is for each client to send and receive data continuously. Since text messages are pretty small and YPIR+SP doesn’t have a lot of overhead, that could be a reasonable way to conceal all metadata, as long as there are not enough people connected to overwhelm the server.