Sorry I am multiplexing them from arduino using mosfets, (highside and low side switching) the brightness does not change much from being constantly powered from the battery or being multiplexed.
Maybe not to your eyes in an indoor environment, but the measured actual light output will
be much lower.
Why would you need to multiplex them? Presumably each segment has its own MOSFET, you can easily control that many MOSFETs from an Arduino using shift registers. This should be the first thing to try, before dismantling it: Change the software to light up one segment continuously, alternating with the same segment lit up multiplexed. Take it outside and see the difference between the two. If it's still not bright enough then you need different LEDs (although finding LEDs that can compete with the sun is a tall order).
PS: If you're going to rebuild it using different LEDs then I suggest using an6884
chips (or ka2284
which work identically) instead of MOSFETs/resistors. They simplify the wiring a lot for jobs like this. (Yes, they're "VU meter" chips but treat them just like a transistor with five current-controlled input legs...all you do is connect the Arduino pin to the input on them and they work great)
With 4 LEDs in series you could control up to 20 LEDs with each one of those. In your case you need one per segment instead of the resistors/MOSFET. Connect the control pins to the outputs of four shift registers, and... job done.