Can't find small supercap, 1mF
You don't need a supercap, you need a few ordinary MLCC capacitors. Looking at Farnell I can find 220 µF X5R, 6.3 volt in 1206 packages, 1.6 mm tall and they are surprisingly cheap (Murata GRM31CR60J227ME11). You can get them with a lower maximum voltage if you don't need that high.

Three of these and you have your capacitance, but add a few extra because X5R is sensitive to DC bias voltages. As you can see, there are room for quite a few, even if you take the courtyard into consideration.