Ham radio is just a synonym for amateur radio. It's not a term that describes the equipment used. This being said, the idea that the equipment needed to perform a moonbounce experiment is somehow outside the realm of possibility for an amateur radio enthusiast is a misconception. The idea that you'd need a dish the size of football field and a small power station to yourself is a blatant lie. Unless you're talking of a
tiny football field (lol) and your idea of a small "power station" is something the size of a portable generator.
There's a list of EME radio enthusiasts/stations
here. Many of them have websites and/or e-mail addresses. Most of them seem to use radio transmitters of no more than 2000W; in fact some claim to use radio transmitters of less than 1000W. Quite a few of them use relatively small antennas or dishes; less than 10m. Some even on the scale of 4-5m.
If someone wants to prove that amateur EME communication is a lie and put one more nail in the coffin of the "round Earth", they might try contacting some of these enthusiasts. They might also try to find an expert in radio communication, someone trustworthy, and go have a look. Talk to those people, see if they have a "legitimate" equipment setup or not. Hell, try to convince them to perform a moon-bounce for you. Who knows, maybe some of these amateurs would welcome a visit. It's worth a shot.
Yeah, I know... silly me. I'm suggesting that the people in charge of the Flat Earth society, or its more prominent members, might actually go out and do some honest-to-god investigating.