I know she is disrespectful toward you, but she does have a child living there and has a right to check what clothing that child has - Whether you like it or not.
In that case - The next time she wants to know, tell your partner to photograph the clothing and send her a picture. That way she will not need to go upstairs. It's likely she forced access to cause an argument between you both. Don't give her the satisfaction of this!
Your partner sounds a bit weak and dim, or he's just playing stupid. He needs to send a strongly worded message/conversation to his Ex, advising she refrains from verbally abusing/harassing you, or you will report it to the police (You can actually do this yourself and show Police the evidence of the abuse you recieve).
Threatening to take the children away is actually illegal now - The actual act of threatening to take them/blackmail, emotional abuse etc is punishable by law. You can refer yourself to social services over this and go for full custody. You can also then apply for supervised visits via mediation only in a contact centre. This way you can block her on social media and only message you both regarding the children.
The fact that his ex continues to message him inappropriately and he has done nothing to stop it, but continues to show you these messages makes me think he does it to make you jealous.
Neither of you sound very trusting of the other with your Ex's, but you must realise they are always going to be a part of your life. For the sake of the children you all need to find a level ground, take the moral high ground, and be firm.
You say you pay for the house? He sounds like a cocklodger. Does he pay for anything? Does he work? Did he pay for anything when married? Why did they divorce?
Maybe show the inappropriate messages to her partner. Forward them on, or screenshot them for social media and make your partner tag her in them and politely ask, "Please stop sexually harassing me. Thank you." Dose of humiliation goes a long way.
Otherwise? Kick him out. You sound too smart for him.