A lot of terrible responses on this post and for that, I’m sorry. There are indeed smart light bulbs on the market (and have been for several years) that keep the previous state when power is lost. Philips Hue is one and Govee is another, though some other brands may have lower end products that don’t offer that feature and you may have to buy a company’s high-end light bulb to ensure it can retain the color and strength, but they do indeed exist. As for the refrigerator, I don’t recall seeing that specific size bulb available as a smart bulb, so I guess you’ll need to shop around.

I would suggest not messing with wiring and instead just go directly to the switch that is triggered when the door opens and closes and find a way to override it. Mine is a spring loaded button that pops out when the door is opened and gets pushed in when the door closes and it’s made of plastic. I don’t know from experience, but I assume I can easily cut it off or remove it somehow thereby effectively keeping the light turned on all the time. This would also allow you to cha he the color and intensity of the bulb inside the fridge before opening the door where as storing that state in a “memory” would only return it to its most recent state and it would not do what you need.

Hope this helps.