Am I unreasonable for how I am handling my daughter being accused of stealing?

I share a daughter with my ex wife. She lives with her mum about half the time. My ex also has a stepdaughter through her husband. The stepdaughter and my daughter do not get along.

Around a month ago the stepdaughter lost her Nintendo switch. She has blamed my daughter for stealing it to get back at her for calling her names. My ex has since searched all over her house and through our daughters things for to switch and has been unable to find it. She is now convinced that our daughter has hidden the switch in her bedroom at my house.

I am unconvinced because my daughter is a really bad liar. She has a lot of tells and breaks down very easily when caught lying. When I asked her about the switch she was very insistant that she does not know what happened to it. I did not see or hear any indication she was lying. So unless my 11 old daughter magically became the best actress in the world within the past couple days I sincerely doubt she stole the switch.

Still my ex is insisting that I let her search through our daughters room at my house or search the room myself. I however find that to be a massive invasion of privacy. I tend to think about it from the point of view of when I was a kid. I would have felt violating to have my things searched through. It’s not even like I hid anything suspicious or wrong I just wouldn’t have wanted my stuff rifled through. Based upon how upset my daughter was after her mom did it at her house I imagine she feels the same. I am of the personal opinion that it is only okay to go through my child’s things if I have reason to believe they could be doing something dangerous or if I have proof beyond shadow of a doubt that they did something wrong. Her stepsister simply saying she stole the switch is not enough proof in my opinion.

Things escalated from there and my ex then gave our daughters switch to her stepsister until hers showed back up. A few days later when I went to pick my daughter up she grabbed the switch and quickly got in my car (I didn’t know about this as she had it in her backpack). When I got home my ex called because they found out my daughter had taken her switch. My ex was irate and demanded I bring it back. After thinking about it I refused to because again she had nothing other than her stepdaughter word as “proof”. Therefore I found her punishment unwarranted. Since I was the one who paid for our daughters switch I saw no reason for my ex to demand it be given to her stepdaughter.

After that my ex started taking away our daughters allowance. We normally take turns giving her her allowance every other week. My ex decided that she would be taking away our daughter allowance on her weeks and giving it to her stepdaughter until she had enough to buy a new one. So in return I decided to give my daughter her double her allowance on my weeks.

My ex now says I am undermining her parenting by making decisions without her, but she is also making decisions without me and I think her punishment is unjust. Especially when her stepdaughter did not receive any punishment for teasing our daughter. Obviously things have gotten out of hand, but I’m not sure what else to do because my ex is absolutely convinced our daughter took the switch and is refusing to budge or talk about it. It’s quite odd because normally we coparent very well and rarely have any issue.