Severe ringing when using high-level gate driver (HIP4081)
Really a well-presented question and your approach is good schematically.  Here are the likely culprits:
First, using MOSFETs like these mean that you have to drive the gates hard to get a fast switching time for two reasons: input capacitance and reverse transfer capacitance.  The driver can deliver amps of current, but not any time soon through the inductive skinny traces you have provided.  Consider what happens: you want the device to switch on fast, and when it does, the drain voltage must drop.  This gate-drain capacitance drives the gate voltage down, turning off the FET, which in turn causes the drain voltage to rise. While the gate drive voltage slowly moves across the switching threshold, it bounces back and forth due to this effect.
Not only must you change the layout to decrease the impedance and provide a current path, but remember that all of the drive current has to come from your bootstrap capacitor, so keep traces short and fat there as well, and you may want to increase the value and get a beefier diode.
Secondly, you have run traces on top of each other on different layers, creating capacitor plates.  You don't want your load to be capacitive, so keep your traces side-by-side, even if on different layers.  These include putting switched traces above the power and ground planes in addition between to the switched voltages themselves.  I often use parallel traces on top of each other on different layers for the same signal with lots of feed-throughs connecting the layers.
Finally, use all of your board to keep the inductances low everywhere you want current to change fast.  Wide traces and big lands work best.
I bet there is an evaluation board for this part - take a peek at how they do the layout and you will see what I mean.
Good luck!