Handshake has an improvement proposal called HIP-0002[1] that utilizes the .well-known directory on your domain's server to enable direct payments. You payment address looks like this: https://<domain>/.well-known/wallets/SYMBOL.
You could even utilize IBAN to send to fiat like so: https://example/.well-known/wallets/USD.
Of course, you'd need do a bit of legwork to implement non-crypto support.
[1]: https://github.com/handshake-org/HIPs/blob/master/HIP-0002.m...