You sound like a wonderful mother and the caution you are exercising here is admirable.
I think it's very easy to say 'see a therapist' but we do need to be realistic - your son made this comment out of the blue one day ago which hardly constitutes evidence of transgender leanings. I think you reaction is absolutely right; provide him with the necessary clothes to experiment since denying him that may well lead to stronger urges in the future. The fact that he removed the items after a couple of minutes to me is an indication that he's not been harbouring a deep seated desire to be female for a long time and it's a passing phase.
You also mentioned that he talked more about it when you put him to bed. His question about the operation may indicate that one of his friends at school had mentioned 'sex changes' and planted the seed in his mind. At 9 years old, he may be starting to notice the prettiness of girls and sees this as a possible route to gaining greater closeness to them rather than a deep seated wish to be one of them.
You are right and your partner is wrong. As a mother, it is your duty to protect and guide your children until they are old enough to stand on their own two feet. Pushing this under the carpet until your son is older will not help and may do a lot of emotional damage to him; as an aside, although my wife knows of my transgender leanings, it is a subject never to be discussed which increases the pressure I already feel from wishing I was someone I'm not. Equally, you need to be careful that you don't lead him down an inappropriate path pushing him to transition when it's just a phase or ignoring his needs when he truly is transgender.
In your position, I think I'd do exactly as you have done, listen to him, answer his questions and let him experiment. If he keeps asking, I think you could ask him if wants to try being a girl for a day, let him dress how he wants and do 'girly' things with him to see how he reacts. That said, it's essential for you to let him take the lead asking him what he wants rather than forcing things onto him. It's completely possible that he will tire of the situation after an hour or two; alternatively if he settles into his female persona and asks you to do it again, then I think you've got a stronger reason to seek professional help.
I hope that, for the sake of both you and your son, it is a passing phase. There are many members of this community who have undergone successful transitions and lead very happy lives but I think all would agree that it was not an easy path to tread. I know from my own point of view that I still don't know for sure where I fit in the transgender spectrum and whilst, throughout my life since childhood, I have gone through phases of wanting to be female regardless of the financial and emotional cost, I have also had times of being perfectly happy as a male. As a teenager, I came very close to telling my mother that I wanted to be a girl but I can see now that this was as a result of envy of their lifestyle rather than a feeling of being trapped in the wrong body. I still have to deal with those feelings some 40 years later and am battling with strong dysphoric feelings as I write this but I do feel that transition would not have been right for me.
With best wishes and hopes that you are able to successfully navigate this issue towards an outcome that will be right for your son, whatever that may be.
Amanda