I have never done anything like that but in looking at them my guess would be yes.

Yes.
Battenberg Lace is a style of FSL.
Just as Crochet Lace is a style of FSL.
EastWitch2

If you go to www.emblibrary.com and do a search for Battenburg Lace, it wil show you a lot of different design packs. Click on any of the pack names and when the next page opens, there is a line near the top of the page that says click on this for special instructions. They provide a downloadable sheet showing you how to make battenburg lace.

advanced-embroidery-designs.com has a lot of battenburg designs
I think all of them there are FSL

If you are wondering if you can stitch it on fabric - you can. But it is made as FSL.