ok,. some rational thought here. She did a naughty thing, but she is 3 and they do that sometimes, because they don't have the control, testing your boundaries etc.
yes they do need to be told it was wrong. either through immediate action, ie. naughty step, take away something they value or positive reinforcement. otherwise we have kids growing up thinking they can do anything they like and causing all sorts of problems.
To be fair, it looks like the OP's partner was trying to be somewhat consistent, by saying that what happens in the home, should also happen elsewhere to avoid mixed messages. I don't necessarily agree with the punishment, but understand the need for consistency too. otherwise does the child end up thinking it is not ok to do x at home, but it is ok to do x outside the home? - just a thought.
I am not convinced that a 3 yr old understand or can vocalise humiliation in this way, as suggested e.g. OMG, everyone else is dressed up, why can't I be? they may ask, but after being told no, they may get upset for a minute or just move onto something else. Perhaps we are projecting a little here? Yes when they are older they probably would feel like this, but i believe that their frontal lobe development doesn't enable them to understand this. Otherwise they wouldn't behave the way they do.
Both my kids would probably forget about it after a time and only remember if I reminded - which probably serves to show that perhaps the punishment is worthless at this age? and a time out or taking away one dress up item that is most coveted is the best appoach.
I personally think the punishment doesn't fit the crime in this instance, but I don't agree that they are too young to understand about discipline, but I am quite strict with both my children.
Think about a couple of things here
1 - new baby brother
2 - new preschool
recipe for misbehaviour.... two big things in her life that mean change and adjustment, so perhaps cut her a little bit of slack.
On the united front thing. DH & I never agree, he always goes to the extreme threat, you will never ever watch a dvd ever. I try positive reinforcement and threats I will carry out, naughty step. Try talking to your DH and have him think through his reactions to discipline and how good his empathy is at a point in time. I have to remind my DH about his approach daily and keep him in check.
Good luck :-)