Note that while in subject-verb agreement some usage manuals say that the final element of an or-coordination should determine the form of the verb, such a rule could not apply with pronoun agreement:
 i # Kim or I may have my application knocked back.
ii Kim or I will have to move my car.
The antecedent for my has to be I, not Kim or I. example [i] is therefore anomalous, and in [ii] there is only one car involved, mine. (CGEL,p.497)
It seems that example [i] needs to be ‘Kim or I may have our applications knocked back.’ From the first line - some usage manuals - do they have to choose was in ‘Kim or I was/were rejected from them’?