Is the output from a 120V 300W UPS enough to kill a person?
SHORT ANSWER: YES! Also be careful handling the circuit board even when its not plugged in. The capacitors can shock you! 

I would bet that the small transformer is there for the small electronics that control the annoying speaker you removed (step down transformer) and the big one is simply for safety and is for the inverter. (Isolation transformer)

LONGER ANSWER: 120 VAC from any wall socket alone will shock you and can kill you. Although there is the debate of "volt vs amps" being more lethal, its known that amperages between .1 and .2 amps (100 or 200 mili-amps) will kill you. That UPS is can put out a whole 2.5 amps which is roughly 100 times more than the lethal amount 

Formula: power = (voltage * amperage)... 

so solving for amperage is...

Amperage =( 300 watts / 120volts ) which is 2.5 amps

The big tansformer is probably an isolation transformer. Its there to keep you from drawing more than 2.5 amps in case of a short. the little one is there to get the voltage down to something suitable for a voltage regulator to manage for the logic circuits that control the UPS. (5 volts maybe)