I think your itinerary is ambitious without kids.

Some thoughts from my recollections of Beijing that i think you may find useful.

Every subway station required you to go through a security checkpoint ( think airport). To reach Tianamen Square (TS) you need to go through an underpass and pass through a similar security checkpoint. I remember thinking TS wasnt particularly interesting, Seeing the Mausoleum would be imo chaotic with kids.

The FC, Beihai part and Temple of Heaven along with just about every other attraction in Beijing is big, very big. None of them are places you can really just pass through in half an hour or so - especially if your daughter really does need to use the toilet every half hour. You will end up skipping alot of the attractions.

I found it very difficult to vsit a few attractions a day due to opening & closing times (along with the size of the places).

I dont recall any food vendors in Beihai -Although they may very well have been food stalls.