The FCC regulations are very long, hard to understand, and occasionally contradictory - I know, I work in the industry. The closest section of the law that comes to mind is for very low power (typically under 1mW) devices that do not require the user to have a licence. These are so called "part 15" devices, which refers to the Code of Federal Regulations (CFR) Title 47, Part 15.
While the user of these sorts of devices doesn't require a licence, the device itself needs to be certified, so that, among other things, it doesn't spew noise all over the RF spectrum. This device wouldn't pass.
There are also experimental licences that can be held by businesses such as RF equipment manufacturers that allow for using equipment that hasn't yet received FCC approval. They are usually band restricted, geographically restricted and power restricted, and you must take care to not have high levels of spurious emissions.
You should - many regulators around the world are modeled after FCC. When I was doing regulatory test work for RFID readers (which were transmitting at 1W ) IIRC Canada, Korea and Japan were very similar to FCC.
Additionally - if you are in Europe - regulations are defined by a body called ETSI (https://www.etsi.org/). For RFID these regs were quite different from FCC - we had to do listen before talk and a variety of other things to ensure compliance.
IIRC you are generally permitted to use whatever frequency you wish within a very limited radius as long as it is for "research purposes"
Edit: this last point is wrong, as mentioned below exemptions seem to apply for certain frequencies with a low powered device https://www.fcc.gov/media/radio/low-power-radio-general-info...