Fonteyne the Kitchen should be fine with a baby, there's a lift which takes you to the restaurant, some tables have room to put a pushchair next to the table. When I went the other week, the place was still reasonably empty just after 12, but after 12.30, it was completely packed, so I hope you are either booking or planning to arrive early.
I'll be eating the children's leftovers (depending on the menu!) tomorrow at work during my 45 minute lunch break, thinking of you all.