My guess is that Mike Morin would be the first called up among the three. He has been a closer since college (and was one of the best in the nation at UNC). Over his entire minor league career he has struck out just 9.3 batters per nine innings while showing impeccable command with 1.9 BB per nine innings (these numbers would be a lot better if it were not for a tough first year at Orem). Take away his first outing at AAA and his numbers would be 2.2 IP 5 SO 1.3 WHIP.
Bedrosian would be a close second to be called up. Bedrosian's numbers are just too hard to ignore. 24.7 K's per nine innings while only walking 2.3 per nine innings. I read a report of his first putting in Arkansas when he struck out all three batters he faced. He put the ball exactly where it needed to be and his fastball was clocked at 93 mph. It's not like he was relying strictly on velocity. He was "pitching".
The least ready would be R.J. Alvarez due to concern over his control problems. But looking at his numbers (though a small sample size) he looks to have vastly improved in that area. He had an awful 5.0 BB per nine innings last year and so far this year it is down to 1.9... Control was the only thing holding Alvarez back from making the big leagues and if he continues to prove that it is a thing of the past, he will be up here fast.
I am no scout as I am just looking at numbers but I hope this is the case that the numbers speak for themselves.
Also there is really no one else in the minors that can help out this pen. Berg has been off to a rough start. Michael Brady (who we got off waivers from the Marlins) just got rocked in his first AAA appearance. Cory Rasmus is their best reliever but he looks like the classic case of heart attack appearance. He strikes out hitters but has walked 7 batters so far in 11 innings.