In hindsight, we're probably not doing ourselves any favors exposure-wise by having absolutely no name on the videos. Lots of people are doing Hatoful LPs, and if I want to find the difference between an outstandingly good one [warning: Hatoful Nageki route spoilers] and an insufferably awful one, those were both easy to pull up in an instant since I just searched for some variant of "(name) hatoful boyfriend." By contrast, if our videos are apparently popular enough to stand out from the crowd of others and warrant TVTropes mention (an honor so far only shared by the legendary Angie Gallant thread and one other person with whom I'm not familiar,) yet our videos have such descriptive titles as "Hatoful Boyfriend Bad Boys Love - 4-1" and "Hatoful Boyfriend Bad Boys Love - 7-3" ....

(I actually did an experiment wherein I tried to find our videos just by searcing for Team Hatoful, as if I were someone who saw the TVTropes article and was intrigued, but since it doesn't have a link, it was up to me to see if I could find them. I ultimately couldn't.)

Of course, that's the ultimate side effect of never having established an official team name! Like you, I always thought that was just an unofficial shorthand nickname, but in the absence of an official alternative, it appears to have stuck.