Running the piezo off 5V won't help directly, since it will just get a swing from 5V-0v (5V) and 5V-3.3V (1.7) so your piezo only gets 3.3V worth of swing, and some model teensies might blow up just for extra fun (3.5 should be 5V tolerent). If you use a transistor drive you can PWM fine and get a 5V swing, and may be easier than trying to get a 6.6V swing by using pins driven in opisition. When you say that you had tried an H bridge, was that using a driver or just trying to have pins going in oppiste polarity? Getting that without bitbanging would be tricky since things like PWM will most likely be starting the on time together so you would need to hack into the PWM code a bit.
The 180 inversion I suspect won't work thinking more about it, since it would require the PWM output to be more complex than I suspect it is. For the basic application you are trying suspect getting a 3.3V piezo (which I think the linked one is intended for) and driving it single ended will get you beeps and such. If it's not loud enough then go for more exotic double ended drives later.