For all we know, the aliens collected Enterprise' hydrogen and then smashed it with a crap-load of antimatter they happened to have on board. Or maybe they used some kind of force field and atom-smashed the hydrogen cloud to create a hell-big fusion reaction.
Or maybe it was just a dubious solution to an otherwise decent episode.
Your last point is the best. The writers wanted something to justify the telepathic contact and came up with an inferior solution. In the episode, the characters discussed why hydrogen fit the "two moons" dream and why the aliens would have mentioned it. They went through a list of chemicals and materials that would react with hydrogen and decided that the aliens wanted hydrogen for that purpose. Anything beyond a chemical reaction was never discussed. So, we've clearly got superior abilities to both the characters and the writers.
The "use the hydrogen as a reactant for antimatter" bit doesn't make a lot of sense because if the aliens have sufficent technology to have antimatter, they have sufficient technology to obtain the most common element in the universe from their own ship.
Any solution we come up with has to deal with the energy nullifying effects of the area. The deflector dish's energy was absorbed, the photon torpedoes on board wouldn't do the trick, so why would putting loads of hydrogen together in the field, in the alien ship, etc with anything else (chemicals, antimatter, bunnies) do anything at all? Unless the aliens are somehow able to extract more energy from a hydrogen + X reaction than Fed tech or manipulate it in a different way unknown to Fed tech, we are left with "it was just a dubious solution to an otherwise decent episode."
Perhaps this was just TNG's "crack in the event horizon" episode.