Stuck on a nanny sharing decision with my disaster of a STBX

First off, I am sorry this is so long. I've been up all night stewing about this and could use your collective perspective.

My 6YO is overweight and I need to help her manage her diet and activity levels. But with week-on/week-off custody, anything I do in my week is undone the next week. 

Hiring a nanny for afternoons could really improve things for my 6YO. The nanny would be charged with making sure she gets at least 30 minutes' strenuous activity each afternoon and could even take her to activities (team sports, etc) that I can't because I'm working. I envision that by the time I got home each day (5-6pm), my girls would have done some serious walking, healthy snacking as appropriate, and have homework done. And also get that one-on-one attention that they deserve. 

Clearly, the nanny is in the best interest of the children. But knowing my X like I do - and now many of you do - I am wondering how making that move could blow up in my face. 

X has wanted to hire a nanny for some time to essentially compensate for his lack of "hands-on parenting" during his weeks. I think he will support this move. But he may try to overreach by trying to boss the nanny about what should be done - overriding my instructions, coming home late, which would increase her cost and her desire to find a new job, etc.  

And there's the question of shared physical custody. I know my STBX is not fit to be the full-time single parent, and he is slowly but surely proving that to third parties (public school and daycare).  Hiring a nanny might compensate for part of his lackluster organizational/parenting skills - handling homework, school paperwork, spotting a fever/infection coming on, etc.  

Hiring a nanny would mean changing the settlement agreement again. Currently it specifically states the care situations for both kids and our contributions thereto. 

People often wonder why it's so hard to just "walk away" or "say no" to an abuser or bully. They just don't realize that the abuser/bully thinks in a way no other human being does. They will constantly astound you with the ways and lengths they will go to to intimidate, threaten, harass, and plain-old thwart the best of your intentions. 

So I guess the bottom line is - Hiring a nanny is best for the kids day-to-day health and emotional well-being. But how much do I trust STBX to share someone who will have access to both our homes? or only my home? Will it make it harder to argue for me to have primary physical custody down the road? 
Will I win the battles but lose the war? Or vice versa?