You could include the hash of the last block in the network as part of your handshake, along with a nonce, and pay out tiny amounts of currency to whoever produ

You could include the hash of the last block in the network as part of your handshake, along with a nonce, and pay out tiny amounts of currency to whoever produces a signed handshake containing that hash (as long as the signed handshake is below a certain number? Not sure if that's useful or not in this situation). But if you want proof of uptime, having a TLS-notarized response with a handshake containing a previous block hash is pretty solid proof.