Some Exchange server users can’t send email from mobile devices

SMTP should be configured to require authentication if you are publicly exposing the ports. Most default configs only allow relaying as authenticated users.

By "outside the org" are you leaving the network? Are you using VPN to connect to 192.168.0.1? mail.yourdomain.com is probably resolving to a public IP address, which would be unreachable while you are inside the network, and would only work outside the firewall if you had your IMAP/SMTP ports open to the Exchange server.

I think that if you were to configure your client to authenticate against both IMAP and SMTP, you would at minimum be working inside your firewall, pointed at 192.168.0.1.

To get it working outside the firewall with a local address, you would need to VPN.

If you don't want to VPN, and want to use mail.yourdomain.com, you could setup the yourdomain.com zone on an internal DNS server (assigned to clients with DHCP) and have it resolve mail.yourdomain.com to 192.168.0.1, and when you are outside the firewall, public DNS would resolve it to the correct public IP.

I finally called Microsoft to get this resolved. The issue appeared to be permissions on the "Client Proxy" HubTransport receive connector. They went into ADSI Edit, Configuration -> Services -> Microsoft Exchange -> DOMAINNAME -> Administrative Groups -> Exchange Administrative Group -> Servers -> SERVERNAME -> Protocols -> SMTP Receive Connectors, then went to the properties for the "Client Proxy SERVERNAME" entry.

Then, on the security tab, went to "Authenticated Users" and made sure "Accept any Sender" and "Accept Authoritative Domain Sender".

Once these were set it began working. I'm not sure what the defaults would be, and if these are the defaults, why ours was not set to that. We didn't change anything with the built in receive connectors.