74LVC inverter gate power consumption
The 5.6K/820&ohm; divider is dropping the BC807's Base voltage to about 630mV, barely enough to turn it on. Your 'speed up' capacitor is doing more than just speeding up the BC807's switching speed - it is actually providing most of the Base drive current. 

The 74LVC1G14 is drawing 'high' current because it has to continuously charge and discharge this large 'speed up' capacitor.  

To reduce inverter current draw you should increase the 820&ohm; resistor to a much higher value (eg. 4k7) so the transistor gets sufficient drive through RB, and reduce the value of the 'speed up' capacitor until it is just sufficient to maintain a fast switching speed (eg. 100pF).