I think it would definitely depend on the exact request, and how strong those beliefs were. Im quite a firm believer in "each to their own" and "Whatever works" providing the child is happy, healthy, secure, cared for, needs met etc..
But the thought of certain practices go against every bone in my body, one in particular is making my stomach churn even now, just thinking about it, so i doubt i could carry through on it.
If it was a preference of mine, and i was asked to do the other, well, fair enough, id probably comply unless i felt it would be detrimental to the childs health or wellbeing.
In saying that though, id tell them upfront so they had the choice to take the offer of baby sitting, or leave it.
But ive also felt the disapointment of someone not following through on your requests because 'their way' is easier...