Primary cell chemistry with the highest volumetric power density
Assumed essential Specs:

Volume: "5mm tall by 10mm diameter"
Power:  150mW power transfer momentary but over a 3 yr span
Type:  Primary cell
Energy:  TBD
Pulse duration: TBD
Temperature: TBD
Budget: TBD  Not critical
Misc. Stress Factors: Ground Benign 

Suggested solution:

2x SR57 Silver Oxide Buttons
Size: 9.5 mmD x 2.55 mmT  ( 5.1mm thick for two) +/-&lt;5%    



Designation: ANSI-1162SO/1165SO, IEC-SR57
Nominal Voltage: 1.55 Volts
Typical Capacity: 51 mAh* (to 1.2 volts)
Capacity Test: 22K ohm continuous drain at 21°C
Typical Weight: 0.8 grams (0.03 oz.)
Typical Volume: 0.19 cubic centimeters (0.012 cubic inch)
Impedance (40 Hz): 10 to 25 ohms    

Assumptions:

two cells ideally 440 watt-seconds (51 mAh * 3600s * 1.2V * 2bat )
    must be derated by load life profile = TBD
    perhaps derated to 1% of ideal with 22 kohm load     


  Average Capacitance in Farads, using E= C (Vi²-Vf²) depends on useful V range
  
  Pd Max load = ½ Pd short cct. or matched impedance to battery string ESR.


est. Short Circuit Power = P=V²/Z = 1.2²/(10 to 25) per cell  x2
est. Pmax = 1/2 of Psc = 1/2 x 280 mW max    


  Pmax = 140 mW @ 1.2V with   Voc = 2.4 V and 20 ohm load
  
  ( This is insufficient but closest match 150 mW )


Datasheet:http://www.microbattery.com/microbat/pdf/energizer/energizer-watch-battery-042-2014.pdf