I had the same problem using Ionic3 to create an application with woocommerce as a backend.
The following code worked for me, but you need to find a way to determine the client ID.
updateCustomer(){
this.customer.id=YOUR_CUSTOMER_ID;
this.customer.password = NEW_PASSWORD;
this.WooCommerce.putAsync("customers/" + this.customer.id, {customer: this.customer}).then((res)=>{
if(JSON.parse(res.body).customer){
console.log('customer updated');
}
}, (err)=>{
console.log(err.body)
})
}