SIM card power supply design, from 3V/50mA into 2.7V/10..100mA
I found this battery which should have enough capacitance and ESR is ≤ 650 mΩ 
http://www.powerstream.com/thin-lithium-ion.htm  

2x12x12.5 mm is too big but there is hope you may find one that fits your package.

But it may have to be current limited for charging so as not to pull down your 50mA supply voltage.