While one can do something even more efficient with pure keybindings, or maybe merely first launching the normal window and then a private one, I thought that it was interesting. AFAIK it can't be done purely on Openbox' menu's functionalities alone.
Code:
#!/bin/bash
xinput query-state 9 2>/dev/null | grep "key\[50\]=down\|key\[62\]=down\|key\[37\]=down\|key\[105\]=down" && incognito="--incognito"
echo ${incognito/--} | aosd_cat
...