Good luck guys, but by the Nine, you should have thought more about the name..
7te and 7tes are going to be confused by a lot of people in the non-German-speaking community which will only make life more difficult not only for those people but for both regiments in question as well.
If the 7te are not willing to come to an agreement with you, as Shifty put it, I'd still argue it's within their right to deny a compromise. The early bird gets the worm, afterall.
Of course, nobody can forbid any of you to carry your respective tags since they are in fact different, but what's the purpose of insisting on your still very similar names even after concern has been expressed and a conflict is apparent? Showing who's the biggest pighead?
You're going to keep the name because "they were unreasonable and impolite", is that course of action particularly reasonable?
I see no benefit in playing defiant in this affair, do what's best for everyone.