It all depends on how many reads overlap If the majority of the reads overlap then there is little that should be gained from treating them as paired end. In fact it should be counterproductive to do so as the system has to deal with more and redundant data.
Logic dictates that providing more information to the system ought to make it perform better. In this case the extra information is that the reads are overlapping and an external tool solved that problem.
Now in reality and practice, the way algorithms are built, tuned and released, depending on the tool and version it just might be that you end up with unexpected performance when choosing one option vs the other. Hence as Manvendra Singh suggests I think it is best to be cautious and evaluate both methods.