Man, Fiscal, that's rough. Sorry to hear it.
In your situation, she may have the "power" by weaponizing custody of your kid, but you have more power, namely, the threat that you can simply vanish and leave her in a permanent economic pit without your income and as a single mother.
To give her a taste of this, you can switch Airbnbs in Lima so she can't show up on your doorstep, ignore all her calls, and spend a week or two without her stressing you. The food is wonderful there, get delivery of all the great stuff that she - as an Argentine - categorically refused to eat. Make pisco cocktails - god knows there's an infinite variety of options. If she has access to a joint bank account, let her keep it (but obviously stop depositing any money in it, beyond the absolute minimum that she needs for support).
Then, after that week or two, you restore contact with a phone call. If she immediately goes ballistic on you during that first call, hang up and cut off contact for another three days. Then try again. Rinse and repeat until she comes to understand - without your ever having mentioned it explicitly - that you could simply dump her and the kid, leaving them without any money or support (aside from your flat, I suppose). Once it sinks in, this will 100% put the fear of god into her, as it would any good funds-seeking Argentine woman, no matter how volatile. She might think you're a cruel sociopath and run you down to anyone that'll listen, but it's a great position from which to bargain.
That'll set it up so that future negotiations are more manageable, and tilted in some basic way in your favor. This basic backstop negotiating position of "if you make this too hard on me, if things don't shake out fairly, if you play the judge against me, whatever it is, then I will just walk away from all of this then you get nothing" should always hang there - without your ever explicitly saying it, especially around witnesses. Even if it's not true, this is your leverage in negotiations, your "power."