Two different years I've had influenza and my daughter never got it. The first time, I took Tamiflu and the second time, I didn't.
Tamiflu does not prevent the flu, it only helps a person get over it a day or so sooner. Our doctor said she does NOT prescribe Tamiflu as a preventative, so my daughter did not take it. We didn't' take any special precautions for her not to get my flu.
In light of my experience, I would not force the 3 year-old to take this medicine.
Here's some info about Tamiflu use in children: http://www.tamiflu.com/hcp/prescribing/hcp_prescribe_chil...
The graph shows that the untreated children got the flu about 17% of the time, those who took the medicine got the flu about 4% of the time.
The daughter who has the flu and is taking the Tamiflu is reducing her chances of passing on the flu to your youngest, also, according to this information.