"Your intention is to take out food, not turn on the light."

Unfortunately good intention only gets you so far. The Gemara says all agree you can't say "oh I just wanted to cut off the chicken's head because my kids like playing with chicken heads; my intention wasn't that the chicken should die!" (This argument is known as psik raisha, or "severed head").

There's some discussion whether it's prohibited if the unintended consequence is one that's irrelevant to you or disadvantageous to you (psik raisha d'lo ichpat lei, psik raisha d'lo nicha lei), but here, having the light in the fridge is helpful (hence the manufacturers put one in!), and when you close the door, having it off is helpful (saves energy, cuts down on a lot of heat that would otherwise be in the fridge.)

Agreed that if someone had to choose (let's say that someone's life depended on taking medicine in the fridge, then putting it back in, three times over shabbos), it would be only one violation to unscrew the light bulb, vs. repeated ones to open and close it.

As Gershon said, the best bet here is to find a non-Jew; depending on the circumstances and the degree of need, you may be able to ask him/her directly, rather than just hint to it.