Sounds like an awesome idea and a logical step for FISD! If I wasn't a man I'd really like for this to come into fruition so I could ask my very personal questions straight out to people with the same fysical experience as me! The FISD has always been about troopers helping troopers so I'd vote for let's make it happen!
I can see all peoples perspective and points in the naming, and hope you can agree on a name that feels good for everyone! I personally kind of don't feel like I could suggest any names (although I think this part of a process like this is always super exciting) since the sub team is not for me specifically! However, I just think I'd come out and say that Vaders Valkyries sounds really cool and has the right "ring" and meaning, like it belongs in the Star Wars universe! I'm not qualified to even guess at this but it feels like people that are not defining as either or could get behind the name too, right?
If you'd want any additional help on maybe graphics or patches or anything like that - I'm at your disposal!