I suspect 'AlexaLog' is only the Log-C curve (1D) so if you're using an official Alexa 3D lut that transforms both the gamma and gamut from Log-C to Rec709 you will get these types of color problems.
Yes, I suspected it was more complex than I thought.
MLVApp uses AlexaLog from
this paper, it is indeed EI800. The ProRes color output is bt609 from ffmpeg. I think Premiere Pro reads it normally by default. The color matrix MLVApp is using came from ACR (actually you're credited in the source code for helping), so it's probably "precise" enough...
Re: WB. Maybe. I didn't do any grading, only set a WB. I'm going only by what I can see on a vector scope and there is no neutral target in the shot. The model has a pink complexion and the beautician/make-up artist is more of an olive color so cooling WB will tend to make the pink hues more blue and less life-like, especially under mixed lighting. If I were grading this I would certainly be using qualifiers to isolate and treat the different skin tones independently.
Yep. Also, the background and the hair tones are in the same shade of grey, so when I try to get the background less magenta the hair just changes together

I'm also using Lumetri from Premiere for this and not Resolve. My fault, I can't expect very much from Lumetri. [Edit: I should just buy a 18% grey card already, I know]
Try doing a basic grade without a lut and see if you still get clipping.
I will.
+3 on mid-tones is quite extreme and yes, will likely cause some banding, especially if done after the lut. Why are you pushing mid-tones so much?
After your last reply I changed it a little, but I was doing it because the skin tones just get's too dark after applying (linear) contrast. I can just apply a general gain, but highlights will clip. I'm using a curve like this (you can see I'm quite agressive in the highlights):

If you have any pro tip for me, I take it

This is purely a subjective observation and you may actually be going for that look but I find the skin smoothing (possibly extreme noise reduction?) in 01.png to be way too much. It completely loses any texture in the skin and looks very unnatural. Try dialing back on the effects and you'll get a much better look 
Thanks. Indeed, the noise reduction (NeatVideo) is way too strong.
Are you using Lumetri in Premier or After Effects?
Premiere Pro.
Thanks a lot for helping Andy, I'm learning very much these days...