You can actually forgo for gyro and here is why. The gyro will only give you changes in rotation rates while the accelerometers will only sense acceleration. I severely doubt that you could turn a car or anything fast enough to harm a pizza, and compensating for this rotation would require the use of a controlled turntable which while awesome would be VERY hard to build.
Instead you just need to compensate for linear accelerations including that due to gravity. With a 3-axis acceleration mounted to the pizza carrying frame one only needs to write code that tries to keep all of the acceleration in the Z-axis. When you are sitting still it should see 1G straight down and no X or Y acceleration. Thus it will orient the sensor and thus the frame so that straight down matches with the direction of gravity. If you accelerate hard the chip will see the 1G of gravity plus the say .5 G of +Y. The combine to form a net force vector that is 63.4 degrees bellow horizontal and it should rotate to shift this net acceleration so it is centered on the Z axis (90 degrees below horizontal) The gyro would be unnecessary to tell you if this acceleration was due to a linear acceleration of the vehicle or a forward tilt of the vehicle. The forward tilt can cause the same acceleration reading by shifting gravity to the Y axis. However because both of these eventualities will affect the pizza identically you do not need to differentiate them.
TLDR:
You don't need a gyro only a 3-axis accelerometer
Write code to keep net acceleration vector aligned with Z-axis (PID might be good here)
You don't need Kalman filters as you won't have gyro data.
It's likely easier than you once thought.
Good luck, sounds like fun!