Thanks for all the replies. I’m measuring 4.05v from the battery pack. Unfortunately the multimeter I scrounged up doesn’t appear to take amperage measurements so I can’t tell if it’s providing enough.
A capacitor sounds like something I should look into; so far I’ve gone from zero knowledge to 4 or 5 broken toys scavenged for LEDs, battery packs and switches and a functioning Beakn.
I’ll have to see if I can find a 4xAA battery pack from some device.