Mother sounds splendid but if she's getting it and they're spending Christmas day there, I wouldn't buy it at this point.
I think buying it and giving it them on Christmas Eve looks like smacking her in the chops deliberately to ruin Christmas day at hers and you don't need that and it doesn't sound like it would be your style.
When is the child's birthday? If it's after Christmas but not too long that would an ideal time for you to buy the second one.
If the birthday is say June and the child has sufficient understanding of the whole dynamic then maybe, maybe give it to him on Boxing Day?
I see it's an big gift for a non-celebration but I might think of doing that.
I'd aim at presents on Boxing Day unless you're hoping to start or have a Christmas Eve tradition. Get them a game (but not as main pressie or they won't have anything they can play with on the day). It shows a good relationship between the people who are supposed to be his parents (even if there isn't one) which is good for their emotional well-being and doesn't spoil the surprise of what they're getting from mother.
Finally, find out if she's buying the switch or the switch lite.
Hopefully it's the lite because you can then get the switch and play up that they have different benefits. The lite is just one player handheld. The switch is the one with the docking station and removable handsets so you can play 2 player games. It also allows you to play online with your mates which would probably be more use at your house than hers.
If it's the switch, not the lite and you do get a second, I would say to her that you felt child would be able to play online with mother in the future with mother using the one at her house. Either that won't happen or it will and it'll be a safe way for them to interact.
Happy Christmas!