Extended Kalman filter - why not use a 2nd order Taylor expansion?
There is nothing wrong with taking a higher order approximation than first order.
An issue in doing so is that you no longer have Gaussian statistics, so some of the standard Kalman filter equations (which are derived using Gaussianity) no longer hold strictly true.
The usual approach to dealing with (otherwise ignored) higher order terms in the standard EKF is to increase either the process noise or the measurement noise variances from what the model tells you they should be. This is like saying,

I know what the noise power is, but I'm going to say the second and higher order terms in my Taylor series expansion add to it so I'm going to increase the noise variance I use in my filter to cater for it.