You can design this easily around four BCD thumbwheel switches, four 7447 decoders (BCD>7seg.) and 7x4=28 transistors with 12 LEDs per segment in an array so both colours use the same voltage with two 3.7V Lipo Batteries. Extras can be added like a charger jack.
(Added) More thought needs to be given to driver voltage and current. Eg. 7446A can drive open collector 30V@40mA which can be two parallel strings of 6 LEDs per segment giving 20.4V for Green and 13V for Red requiring 2 boost converters slightly higher, but then saving 28 transistors.
12x7x2 =168 pairs of RED(2.2V) & GREEN (3.2V) LEDs with 4x7seg = 28 (Bipolar or MOSFET) switches with selected voltages or current sources. YOu can use a 2x8bit latch if you wish to update the display instantly rather than switch encoded in real-time as you wish. Remember to filter the switches.
For 5mm LEDs consider > Power, Beam Angle, Battery voltage, SMPS, elliptical 5mm Lens diffused or clear or 5mm round or? I would use 34x70deg wide elliptical.
POWER 25W total if all segments 88 88 are on.
Red you need at least 168 x2.2V x 20mA = 8.4W est.
Green for each LED in a segment in series 168x3.4V x20mA = 11.4W
For voltage, I would choose a common denominator voltage for each segment.
e.g. 2 Green is close to 3 Red in Voltage. 2x3.4V=6.8V(grn) ~ 3x2.2V=6.6 (red)
Therefore consider each segment as an array of 4P3S (Red) and 3P4S (green) for an average voltage of 6.7V which is easily derived from 2 LiPo batteries in series.
Max current with "8888" would be:
Red 4P x7seg x2dig x 0.02A =1.12A
Grn 3P x7seg x2dig x 0.02A=0.084A
Hammer driver Darlington's have 7 drivers in a chip could be used or very cheap SMT MOSFETS with 35mOhm resistance.
Better if you use low Vsat or RdsOn to use all of the 7.2V battery range when depleted to 6V. (Red can be obtained down to 2.0Vx3S=6V )
LiPo power should last many games using 7.2V battery with a ~3 Ohm series current limiter for each segment or use a Polyfuse for ~100mA.
Much more thought in making it robust, high efficiency, wide viewing angle and variable brightness can be added.
Much more thought into high quality Elliptical diffused LEDs with over 5000 mcd will be suggested from a good supplier I know.