Are the rules different around the world? Is a technicians class amateur radio license sufficient? What about the requirement to transmit your call sign every 10 minutes? Is that necessary in this use case and if so, how do you comply?
Addressing this specifically, just in case you decide to get your ticket for the purpose of flying (and many people do!). Yes, things are different around the world, although there are also many commonalities. I will only discuss the US, since that's what I'm familiar with.
First, a Technician license in the US grants all privileges on amateur frequencies above 50MHz. As long as what you want to do is VHF/UHF/microwave, Tech will let you do it.
Second, the rules allow several different ways to transmit a callsign, depending on the type of signal. For NTSC television signals, the callsign can be included as part of the image. In other words, if you can program your FPV transmitter to overlay your callsign permanently in the corner of the image, that satisfies the ID requirement.
There is also an exemption to the ID requirements for "telecommand of model craft" (that'd be your flight controller) . If you attach a sticker with your callsign, name, and address to the transmitter (and if it's otherwise technically compliant) you can use it on the ham bands up to 1 watt.