Since the current consumption of your smart bulb is only 7 watts, I am assuming it must be an LED bulb. Internally, most LED bulbs use a power supply design consisting of a "dropper" capacitor, full bridge rectifier, and a smoothing capacitor.
The "dropper" capacitor is wired inline with the hot from the mains. The voltage from the mains is 120 VAC at 60 Hz. The voltage is higher than what is required. This capacitor is used to "drop" the voltage down, since the capacitor is an impedance in the AC circuit. This increases the component cost, but is very efficient since the capacitor has very little loss at 60 Hz.
The full bridge rectifier creates DC current from the AC current. I won't elaborate on this since there is plenty of reference material on the internet.
The smoothing capacitor causes the DC current side to have less ripple. The ripple would be 100% of the peak voltage without this capacitor. On most bulbs this is just there so the LEDs do not visibly blink to the human eye. Since this is a smart bulb, the smoothing capacitor is probably better than average to allow the electronics to function correctly.
Such a power supply configuration is non-isolated and potentially lethal. But since it is contained in the package of the bulb, it is relatively safe.
In order to act as a very brief UPS, you could modify the bulb to use a much higher capacity smoothing capacitor. The capacitor's energy storage is measured in farads, you would want a larger capacitor. But a capacitor is a poor UPS. The voltage drops linearly with respect to charge, unlike a battery's voltage. So there a few problems:
- By the time you select a large enough capacitor, it will no longer fit inside the housing of the bulb.
- When the bulb is first powered on, the capacitor must be charged. The full bridge is responsible for carrying the current for this. The capacitor draws an enormous amount of current initially. This could damage the full bridge rectifier by exceeding it's current rating.
If you own your home and can afford it, a much better option would be to have the lighting in your home powered by a separate electrical circuit. This circuit can then include backup power for your lighting. The equipment for this is commonplace in commercial settings due to fire codes. It would not require any modifications to your bulbs. This actually makes your home significantly safer as well.