My 12 year old daughter will probably spend most of the day in shul on R”H. Is it correct to ask her to stay home part of the time to babysit the younger children so I also get a chance to go to shul? Or is it more of a priority for her to daven than for me (since the obligation to watch the children is really mine and not hers)?
It is correct to ask her to stay home part of the time so you can also get a chance to daven in shul.
Although watching the children may not be her obligation, she also has no obligation to daven specifically in shul (except to hear shofar). She does have an obligation to make sure that her mother also gets a chance to daven, as part of kibud av v’em. When she davens in shul, she will be able to answer amen etc, and feel the davening better. However when she babysits at home so her mother can daven she is doing mitzvas kibud av v’em which is d’orayso v’ahavta l’reiacha k’mocha which is also d’orayso, chesed, which is d’rabonon, and another important lesson… to take part of the family responsibilities, and to be considerate of her mother!
Therefore this is what H-shem would want her to do, to switch off with you so you can also daven in shul.