My DH had a similar-ish situation as a child. Not as challenging as what your DC is facing in lots of ways though.
His father also didn't allow him to take items home, despite that being where he spent 99% of his time. There was a thing over bikes one year. His dad made a big show of buying this fancy bike but then wouldn't let him take it to his mum's house, so his mum also bought one. Otherwise he wouldn't have had a bike except for a couple of Saturdays a month.
DH will always say that he's very grateful to his mum for letting him figure out what a twat his dad is by himself. That's not to say she didn't try and protect him where she could, but there were lots of times when she chose not to dig her heels in, or chose to avoid a standoff, because she knew that to stand her ground (however justified she would have been) would have placed DH at the centre of a conflict and would have given ammunition to his father. Her desire to "win" the fight with her exH was secondary to her desire to make things easier and calmer for her son. Whatever the situation, DH loved his dad as a child. He would have been deeply distressed to think of his dad feeling disappointed or hurt, even though as an adult he can see that this was a very unhealthy dynamic and his dad was just being manipulative and using these kind of situations to try and garner favour with DH and undermine his relationship with his mum. Shitbag.
So I think in your situation, I would let mum have her moment and give the console on Christmas day. I would move my "Christmas day" to Boxing Day and give the duplicate console then. There's potentially going to be a short period of disappointment for DC where they realise they can't bring the console home, but that will be quickly rectified when you do your gifts on Boxing Day.
If you sneak in and give the console first on Christmas Eve, you will create a whole load of conflict with DC's mother who will likely drip poison in their ear. It's possible that this will be extremely upsetting for DC and lead to crushing guilt - guilt that their mum is upset, guilt that bad things are being said about you. The gift will be ruined. It's still his/her mum and it's a complex situation. If you act with grace here, your DC will understand for themselves one day and will see the situation for what it is.