Braintree is not a first party payment gateway and I have never used it so I can't say for sure but from a quick look through the gateway files on GitHub I do not see anything that would support how Braintree adds PayPal support from their API guidelines.
Having said that you could easily add it if you are are comfortable with PHP or you can use the built in PayPal Express gateway along with the Braintree gateway to allow checkout with either PayPal or Braintree. If you were to go the Store route you would need to use the add {field:payment_method}
to your template to allow the user to select either Braintree or PayPal. Or if you were looking to customize it a bit more you could roll your html something like this:
How would you like to pay for your order?
<label><input type="radio" name="payment_method" value="manual"> Cheque / Bank Deposit</label>
<label><input type="radio" name="payment_method" value="paypal"> Paypal</label>
<label><input type="radio" name="payment_method" value="authorize_net"> Credit Card</label>
<!-- optionally you could use javascript to display a credit card form here when necessary -->
Full documentation for the template tags is here and setting up payment methods is here