I think the following link may help you:
http://o365info.com/manual-configuration-of-outlook-mail/
Use Microsoft Remote Connectivity Analyzer to find a specific server name to use instead of outlook.office365.com
This worked for me in Office 2013 on a PC for Office 365 with a custom domain name where autodiscover would not work. I also had to use Anonymous authentication and Outlook Anywhere settings to connect to Microsoft Exchange using HTTP... so I am not sure how those steps can be done in Touchdown. But I do think finding your specific server (similar to the format "7d30b5ec-4218-4f7b-a266-55f45f45960a@o365info.com") may be helpful.
I came on this forum to find out how to add a second account (Office 365) to Touchdown, but from the other thread it looks like multiple accounts cannot be polling at the same time, so I will use Outlook for Android for my second account. Therefore I will not personally be trying these steps in Touchdown, but hopefully I have pointed you in a useful direction.