Are wrist straps or hooks a possibility?
I ask, you said that the injury occurred in the area of the phalanges (fingers) rather than the metacarpals (which end in the knuckles, making up a large part of the upper palm).
Such an injury would allow for pressing exercises, as well as straight arm pull downs (activating the lats) and pulling movements with the use of straps or (if articulation of the finger is not possible) the cautious use of hooks.
Unless there is some damage to the area of the metacarpals or lower on the hand, I cannot imagine that this would not be an option, especially with lighter weights.
That being said, I do not see this as a major reason to discontinue the cycle.