Li Ning Saga or Saga TD:
https://smile.amazon.com/gp/product/B074WWBG12/ref=ppx_od_dt_b_asin_title_o00_s00?ie=UTF8&psc=1
Narrow, true to length, excellent heel cup, made for lunging. Very comfortable, good quality, soles hold up well on non-metallic strips (haven't used them extensively on metallic strips).
These are high-level badminton shoes. Good deal for under $100. Added bonus: amazingly garish colors available. I mean, wow.
Asics actually fit me better since I do not have especially narrow feet, but if you do these are an excellent option.