I already understand from the link posted that you can bring batteries – I certainly looked there before I posted here. But it sounds like they're saying you can bring batteries not necessarily the items that they are powered by. the second paragraph is clear about bringing items powered by larger batteries, but the first paragraph makes it sound like they're talking about freestanding batteries.

For the flyswatter I've now gotten some more information from my research that it really is up to the individual TSA employee – some consider them potential weapons and some don't care. Which would tell me that an item powered by a lithium-ion battery would be okay. But neither of the items indicate the wattage or other information clearly.