The pectoralis minor has it origins at the third, forth, and fifth ribs—just lateral to the costal cartilages—and it inserts on the coracoid process of the scapula. It acts to depress and stabilise the shoulder girdle, being functionally equivalent to the inferior fibres of the trapezius. Importantly, it does this independently of the position and orientation of the humerus, and it is therefore recruited in any motion that involves shoulder depression.
The question therefore becomes, which exercises, other than the straight-arm dip, can effectively load shoulder depression?
Of course, there are any number of contrivances that creative trainers have come up with to perform this movement. But if the dip is the king of upper-body, body-weight exercises, then the pull-up is surely the queen. (Or should that be the other way around?) And it can easily be modified—again, by maintaining straight arms—to isolate depression. Hang, elevate, hold, control depression. Simple.
As an addendum, it may be possible to address the problem with your wrists in the dip. Wrist soreness in such exercises is typically the consequence of excessive pressure being applied to an extended joint. I recommend trying a thumbless grip—that is, a grip in which the thumb is brought around to the same side as the fingers. This enlarges the contact area, thereby reducing pressure on the palm (pressure = force ÷ area), and allowing the wrist to straighten such that the line of force acts directly through the forearm without forcing the wrist further into extension.
I hope that is helpful.