I don't think that's something available on Stripe, but there are a few tools available in the payments processing ecosystem that allows you to do just that: "notifications of fraud" and "requests for information".
Basically, the first is a notification triggered during the chargeback initialization process on the bank, but before the chargeback is actually processed. So when you receive the notification, you can still issue refunds, cancelling out the incoming chargeback.
The second is similar, but is at an even earlier step of the chargeback process where the user simply asks its bank some information about a line on their statement they don't recognize.
You're 100% right, what Braintree does can be done using PayPal's API directly, even what Braintree calls their "in-context" PayPal payments. This is pretty much exactly what we've done at https://processout.com (shameless plug!)
Shameless plug here! We also created something we called our Smart Router engine at https://processout.com, which helps you connect to dozens of payment providers and optimize your transactions.
Basically, the first is a notification triggered during the chargeback initialization process on the bank, but before the chargeback is actually processed. So when you receive the notification, you can still issue refunds, cancelling out the incoming chargeback.
The second is similar, but is at an even earlier step of the chargeback process where the user simply asks its bank some information about a line on their statement they don't recognize.