I'm a fan of the Delta Flyer. Two bridge stations, so you're not stuck with the longer cooldown on your second power like you are with the runabout's tractor beam or the Peregrine's rapid fire (though just having rapid fire is nice).
The Tal'kyr is an interesting one. I've played with it on Tribble but don't have it on live. Two bridge stations plus a built in power, but it's "only" as fast as an escort, which by shuttle standards might as well be a Galaxy-X stuck in a tractor beam.
The "best" would probably be the Delta-class Shuttle (Delta Flyer) since you have an Ensign Tactical slot and an Ensign Universal slot, whereas you only have one Ensign Universal slot in all the other options (except for the Tal'Kyr Support craft, which has two Ensign Science slots). STO Wiki has a nice comparison table here for all of the Shuttles (including the KDF ones in the same table).
That said, I've only completed the mission so far in a Runabout, and the only problems I had while in the Runabout were of my own making.
Edited to add: I see I was beaten twice while typing!
I'd generally call it a toss-up between the Delta and the Yacht. Extra equipment slot on the Yacht vs. slightly higher maneuverability and stronger shields on the Delta. (I usually take EPtS as one BOff power, so the stations aren't an issue for me.)