In my opinion addition of TVS diode and a capacitor between G and E would be good. This TVS diode will suppress and voltage as well as the current transients. The cap can handle smaller transients and also act as a smoothing capacitor with the TVS. Again, I am referring some links here for you so that you can take an informed call on how to implement the protection circuit.
Maybe gate to source capacitance act as smoothing but however the bootstrap capacitor do the same task actually it connects between your bridge's emmitter(or source when using nmos) and gate.
I've just finished my 3 phase motor driver circuit. I want to help you when seeing your topic cause i was struggle with the bootstrap capacitor.
Firstly, when you decide to drive your fets or igbts with using bootstrap method, you actually can not reach at %100 duty theoratically because, when %100 duty signal act as dc so it can not charge your bootstrap capacitor properly. I writed a matlab code for calculating bootstrap capacitor minimum value in related to almost every parameter even bootstrap capacitor leakeage current. I changed its parameters suited with your igbt's total gate charge and frequency etc. Eventually, your required value for bootstrap capacitor is approximately minimum
0.7uF. However, in my opinion you should use 2.2uF to 4.7uF electrolytic capacitor which voltage rating is above your supply voltage of gate driver which is 15V. Also, you should use 0.1nF ceramic capacitor parallel to bootstrap for filtering high frequency components.
Also, you should measure the capacitor charging current which should be desirable level in related to your frequency. Maybe, you have to use bootstrap resistor to limit your charging current whose value is a few ohms.
The other thing you should consider is that you should put a resistor whose value is 1k or maybe higher between gate to emmitter for high switch igbt and gate to ground for lower one. It is used for decharging the gate current in the case of unwanted gate charge while starting-up.
If your gate drivers output current is not enough to drive igbt' gate. You should use totem-pole structure in front of the output to feed the current.
I am sending to you my matlab code maybe you want to change any variables to see the variation of bootstrap capacitor value.
Hope it helps.