Is there a problem with making a baby do a Melacha on Shabbos for himself [closed]

My baby sleeps in a cot with a Cot Mobile playing on top of him. This is a toy that basically has a merry go round that sings, and my baby looks at it and enjoys it and is therefore not bored. On Shabbos, I find it very hard that my baby has nothing entertaining to look at because he cries a lot when he is bored. The Cot mobile goes off after 15 minutes of play.

May I hold my baby close to the on button and let his legs kick the button making the music go on or is there a problem?

If you would position the Cot Mobile in a way that you would not need to hold him close to the on button, but rather lay him down and he could reach it himself then it would be no problem. However holding him in position would seem to be an extension of your hands.

I can't find the source now but IIRC one is required to allow the members of his household to rest on Shabbos, even those that are not required to keep Shabbos on their own.

I have since asked Rabbi Gershon Meltzer from Yerusholayim this question. I posited him the question in this form and in a case of a child switching on a light.

He told me that there are those who are lenient about this because of a Shaarei Tzion Simon תר"מ that says one may put food in front of a child outside of the Succah. This seems to apply that there is no problem with your child doing an Averah. However, he thinks that it is different here.

There, you are not doing a Maaseh Averah. But by holding a child, it could very well be that a Maaseh done by a child who is held, is considered to be done by the holder. He said that Rabbi Elyashiv zt"l was against doing such a thing and therefore he is reluctant to be Mattir it.