Even if your studs are 15" apart, and the holes are 16" apart, there will be a portion of the stud behind the mounting holes, just not the center of the studs.
Angling the lag screws a little, and using washers to buffer the head of the lags against the metal of the rack will aid in tightening.
Beyond that, I would not hesitate to elongate the existing hole to allow it to get a better centering on the studs, or even abandoning using lags, and drilling a series of 3/16" holes, maybe 4 or 5 or more and using drywall screws on each side instead. There has been mention that drywall screws don't hold a lot, since they are brittle, but if there is enough used, brittle don't matter. In a pinch, in my opinion, 4 drywall screws would hold 200 lbs. but I would not stop there, I would use at least that many on just one side, if I used the screws.
The screws would not have a tendency so much as to split out the side of the stud, compared to lags. But a properly sized pilot hole will allow the lags to perform in this "tight" situation.
Angling any of these fasteners will help deter the splitting out at the side of the stud.