First off, I haven't read completely through this thread -- I started it then had to go elsewhere, which gave me time to think about the premise. I decided to go ahead and post my thoughts before reading everything in the hopes of interjecting something new (brainstorming and all that). My apologies if I parallel, contradict, or otherwise derail what you all have generated the past few days.
I love the idea of an "official" what-Catapult-should-have-been variant. I've tried to keep in mind the summary that Carcassonne is a light tile-laying game and not a game of dexterity. So here are my rough thoughts:
The Tiles: Fairs attract people. When a fair tile is played, meeple will migrate to it ala Wagons. From how far and how many? I would like to allow stacking meeples on this tile, it gives it a unique flavor. Are the meeples stuck here for the game? You've robbed other features of their meeples, are the losing players somehow compensated? That is, is the tile an aggressive point-robber or should it be painted as a positive, happy play option.
The Tokens: I see these as bonus play tiles, like the Gold Nugget from H&G or Kevin Graham's variant. At what point in the game they come into play is undecided - specific scoring points? When a fair tile is played? Seduction - trade positions with another player's meeple. Knock Out - reclaim a "stuck" meeple, like farmers or incomplete knights. Target & Catch - no idea.
The Catapult: using this element as something other than a dexterity piece is the biggest challenge. The only real option I see is to use it as a randomizer. Perhaps meeples trapped on the fair tile can escape only when they "ride the catapult" (no idea what that means. heh).
Then on my morning commute, the above points coelesced into something like this:
When a fair tile is played, it attracts meeples and scores. Then the bonus tile round is played out, which goes something like this:
Beginning with the player who has the lowest score (I like to give the underdog a bone every now and then), each player selects one of their tokens to play this round. In clockwise order, each player choses one, but they can not repeat what another has chosen. (I know that will easily leave some players out, but deal with it. They will get their chance soon enough.) Players may always chose to pass in a catapult round.
The selected tokens are stacked on the catapult, all together, and flung. The lowest score player gets to use his bonus token, that's a given. The token which lands closest to that player's token (this is where the measuring bar comes in) also gets to use his. The other players reclaim their tokens for later catapult rounds. Used tokens are out of play for the duration of the game.
That's my very rough first pass. Now that I've muddied the waters, off I go to see what else has developed.