Getting more current through a BJT
Coin cells or no coin cells, the base resistors R3 and R4 are much to big. You will need to chose transistors that can sink more current and feed them with a base current of 15mA or so (divide your desired LED current by 50, typically). Your base resistors need to be a few hundred ohms.

Also, you shouldn't use a PNP as a low side switch. It will only act as an emitter follower and never reach saturation. Move the PNP to the high side...that is, between the anode of D2 and the supply voltage.