RSG, I don't think talking to her about a haircut is showing weakness, but I think her whining about you giving him a haircut and agreeing with her that you shouldn't have done it, and won't be doing it again shows too much weakness.
Maybe saying something like "I understand your upset about me getting S haircut, but since you left family I will be making family decisions on my own, but will consider discussing with you in the future when I feel it is appropriate"..
I'm just trying to say that she has to understand that while she is away from your family you are a single father and will be living your life as such.
I'm just shooting from the hip here, so not real sure I'm giving the right perspective, but think it's at least worth considering. Maybe writing down what her leaving has cost her (as far as family stuff) and what hasn't changed would help to see if she's actually facing any consequences for leaving and having A.
M - 9 1/2 years 5/5/16 - Bomb drop - 3 week EA 10/31/16 - We sold house 01/10/18 - D Finalized