how hard is it to get a sorority bid?

Zeta-Difficult: probably should be your main focus of the week if you want a bid, you should walk the walk and talk the talk
EDM-Moderate: come to their events, be girly and don't be a jerk and you'll probably get a bid
Theta-Moderate/Difficult: you should really show vested interest in order to get a bid
AST- Moderate/Difficult: show them you could fit with them and don't be a jerk and you'll probably get a bid

AST doesn't have to recruit to a campus total, that's not how national sororities work. At least at Kenyon, because they're the only national organization they play by the same rules as the local groups. If there was a second national sorority they would have to reach an agreement on how many new members they're allowed to recruit, but since that's not the case, they're on their own. Additionally, don't just assume you'll get in. I've had a number of friends who didn't receive bids from AST and instead joined EDM.

I don't think anybody is trying to negatively pit sororities against each other; getting into one organization and not the other suggests very little about either of them, it happens all the time.

Things may have changed, but AST's initial process was to extend bids to girls who didn't get into the other organizations and Delta Gamma was dropped at Kenyon in '98 or so because they couldn't keep their numbers up to national standards. Look up "quota; it is, in fact, a standard schools with national orgs use for member recruitment. I'm sure things settled now that they've gotten more established, but this is not suggestive of them as a group of people, but the reality of being national- keeping numbers up.