I tried to use the web client on the Fire TV stick, both in Silk and Firefox. The problem is that the Fire TV browsers' copy and paste is not fully implemented (you can copy, but you can't paste).You can manually type the code in, but then there is no ENTER key to submit the page. A couple of buttons would solve the problem...
Bump...
I'm confused why you're ending up on this page. If you visit my.channelsdvr.net it should not show this page at all.
Do you have remote access enabled on both dvrs?
It appears that my attempt to connect the Channels app on the stick to a remote DVR, which invoked the new 5 digit code process, also caused the web client authentication to automatically complete. As I was watching, I saw the url for the interstitial page briefly appear in the message area.