Can't reach local domain when configuring SOHO to use DHCP in DMZ

I’m currently using Bell Fibe business internet with my SOHO router using PPOE authentication which works fine - I can put in my own while inside my LAN and get to everything just fine. The Bell Fibe modem is configured in Bridge mode in this case. However, I want to add a RATtrap device for security between the modem and my SOHO router, but it will not work. The folks at RATtrap suggested I setup the Bell Fibe modem to perform the PPOE authentication and place my router in the DMZ while configuring the SOHO to just use DHCP. From the outside this works fine. I tested the SMTP and I can send email from the outside to my mail server without issue. I can go to the domain with a browser (from the outside) and that works great as well. However, I cannot get to my domain by either Outlook client or browse using the DNS name. Are the any network gurus that can suggest a solution?

Thanks in advance