Please check that multimeter. A single cell li-ion should not go as high as 4.5V! That would be very near to cause thermal runaway.
Not sure about the 3DS, but the DSi won't allow turning on with that high voltage (I've connected a lab power supply to the power input). I believe it accepted up to 4.3V.
Can't simply test this with 3DS as it needs a thermistor connected at the battery terminals.
Next it would be nice to know what happens when pushing (power) button.
- Nothing? → ???
- Blue LED on, fading off → Modded console not finding boot file
- Blue LED on, abruptly off (sometimes with popping noise) → vital component not connect or ribbon cable to the upper screen damaged
Additional information please:
- Drop damage?
- Water damage?
- Anything else?
Good luck!