Ways to power my circuit during brief power outages to retain switch state?
The most straightforward way to solve this problem is to use a latching relay, and configuring your TTP233 to give a momentary, rather than a latched output.
A latching relay works on a mechanical latching principle, and will remember it's state even through a power outage. To change the state of a latching relay, one applies a momentary current, which can be provided via the TTP233.
An electronic replacement for a latching relay is possible, but seems unnecessarily complex for what you are trying to achieve.