At the hospital where I first started OB, they didn't use any betadine for deliveries. We prepped with hibiclens. I never thought about the betadine for caths after deliveries though. I always use the betadine in the kit for caths. I have never really had anyone complain, usually because the reason they can't void is because they are still numb from the epidural.
At my current hospital, our docs prep with betadine, and if I am cathing them directly after delivery, I just use a red rubber cath, as I figure they have already been cleaned during the delivery prep.
Like I said, if they can't go because they are still numb from the epidural, then they really don't have any discomfort from the betadine, or the cath itself. Usually, if they are edematous enough after delivery that it looks like they will have trouble, our docs order a foley overnight, or few a few hours, until the swelling goes down. Then we just insert the foley while they are still in delivery position before the area gets contaminated, so I don't use the betadine then.
I wouldn't think that one time use of betadine to cath would be too detrimental to any perineal lacerations or episiotomies.
I actually usually clean them up after delivery by using the leftover betadine solution (mixed with SW) from delivery, followed by the sterile water.
I personally don't see the point in the whole prep before delivery and all the "sterile technique" involved, as a vag delivery is anything but sterile. I can see cleaning the site before suturing, but otherwise, I dont see the point.
The last place I worked, we had one doc that had us do the hibiclens prep, but she didn't really care if the table was "contaminated" because she was also of the belief that vag deliveries aren't sterile in the first place.