Hi,
HTTP traffic can be used to access the SBG control centre if configured, however this is not your problem here.
The reason you can't get to OWA is outlined in the error message you've quoted:
"I get an error because it says I can't have two NAT entries that originate from a single IP (my public mail.mycompany.com IP) NAT'd to two different internal IPs."
It's your firewall/router that is the problem here.
Previously, when one server was handling the SMTP and HTTP/S (I hope it's TLS'd anyway) it was fine, because your Exchange server was handling all of that.
Now, SBG does not proxy nor route HTTP traffic, so you need to set your firewall/router to NAT the inbound HTTP requests to Exchange.
It seems that your firewall won't let you because you are doing routing at the hostname level.
You have two options:
1. Create a new A record (i.e smtp.yourdomain.com) and use that as your primary MX. Redirect that host to SBG and redirect mail.yourdomain.com to exchange.
2. Work out how to port forward 80/443 to exchange and 25 to SBG at the protocol level not the host level.
Hope that helps, let me know if you need further details.
//ian